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Diffusion and Peer Influence
“you are who you associate with”
Time
Diffusion & Peer Influence
1. Diffusion
A. Compartmental Models
B. Network Diffusion
i. Topology
ii. Timing
iii. Structural Transmission
a. Complex contagion
2. Peer Influence
Coleman, Katz and Menzel,
“Diffusion of an innovation among
physicians” Sociometry (1957)
Our substantive interest in networks
is often in how things move through
them, from disease to ideas to
behavior.
Network Diffusion & Peer Influence
Basics
Figure 2. Binge Drinking
Predicted Probabilities by
Gender and Friends’ Prior
Drinking.
Derek A. Kreager, and Dana L. Haynie American
Sociological Review 2011;76:737-763
Romantic partnerships
in high school lead to
adoption of partners’
friends’ behaviors
Network Diffusion & Peer Influence
Basics
Table 4. Proportion of Ties Created or Maintained Over Time by Network Process and
Depression Level.
David R. Schaefer et al. American Sociological Review
2011;76:764-785
Health effects on ties
Depressed students form ties through non-normative network processes
Network Diffusion & Peer Influence
Basics
Classic (disease) diffusion makes use of compartmental models. Large N and homogenous
mixing allows one to express spread as generalized probability models.
Works very well for highly infectious bits in large populations…
SI(S) model – actors are in only two
states, susceptible or infectious.
See: https://wiki.eclipse.org/Introduction_to_Compartment_Models for general introduction.
SIIR(S) model – adds an “exposed” but
not infectious state and recovered.
Network Diffusion & Peer Influence
Basics – might even help understand the zombie apocalypse
http://loe.org/images/content/091023/Zombie%20Publication.pdf
Network Diffusion & Peer Influence
Basics
Network Models
Same basic SI(R,Z,etc) setup, but connectivity is not assumed random, rather it is structured by
the network contact pattern.
If pij is small or the network is very clustered, these two can yield very different diffusion
patterns.*
Real Random
*these conditions do matter. Compartmental models work surprisingly well if the network is large, dense or the bit highly
infectiousness…because most networks have a bit of randomness in them. We are focusing on the elements that are unique/different for
network as opposed to general diffusion.
Network Diffusion & Peer Influence
Basics
If 0 < pij < 1
Network Diffusion & Peer Influence
Basics
If 0 < pij < 1
0.01
0.06
0.11
0.26
0.46
In addition to* the dyadic probability that one actor passes something to another
(pij), two factors affect flow through a network:
Topology
- the shape, or form, of the network
- Example: one actor cannot pass information to another unless they are
either directly or indirectly connected
Time
- the timing of contact matters
- Example: an actor cannot pass information he has not receive yet
*This is a big conditional! – lots of work on how the dyadic transmission rate may differ across
populations.
Key Question: What features of a network contribute most to diffusion potential?
Network Diffusion & Peer Influence
Network diffusion features
Use simulation tools to explore the relative effects of structural
connectivity features
• A network has to be connected for a bit to pass over it
• If transmission is uncertain, the longer the distance the lower the likelihood of spread.
0
0.2
0.4
2 3 4 5 6
Path distance
probability
Distance and diffusion (p(transfer)=pij
dist
Here pij of 0.6
Network Diffusion & Peer Influence
Network diffusion features
We need:
(1) reachability
(2) distance
(3) local clustering
(4) multiple routes
(5) star spreaders
• Local clustering turns flow “in” on a potential transmission tree
Arcs: 11
Largest component: 12,
Clustering: 0
Arcs: 11
Largest component: 8,
Clustering: 0.205
We need:
(1) reachability
(2) distance
(3) local clustering
(4) multiple routes
(5) star spreaders
Network Diffusion & Peer Influence
Network diffusion features
• The more alternate routes one has for transmission, the more likely flow should be.
• Operationalize alternate routes with structural cohesion
We need:
(1) reachability
(2) distance
(3) local clustering
(4) multiple routes
(5) star spreaders
Network Diffusion & Peer Influence
Network diffusion features
Probability of transfer
by distance and number of non-overlapping paths, assume a constant pij of 0.6
0
0.2
0.4
0.6
0.8
1
1.2
2 3 4 5 6
Path distance
probability
1 path
C
P
X Y
10 paths
5 paths
2 paths
Cohesion  Redundancy Diffusion
Network Diffusion & Peer Influence
Network diffusion features
0 1 2 3
Node Connectivity
As number of node-independent paths
C
P
X Y
Structural Cohesion:
A network’s structural cohesion is equal to the minimum number of
actors who, if removed from the network, would disconnect it.
Network Diffusion & Peer Influence
Network diffusion features
STD Transmission danger: sex or drugs?
Structural core more realistic than nominal core
C
P
X Y
Data from “Project 90,” of a high-risk population in Colorado Springs
Network Diffusion & Peer Influence
Network diffusion features
• Much of the work on “core groups” or “at risk” populations focus on high-degree nodes. The
assumption is that high-degree nodes are likely to contact lots of people.
We need:
(1) reachability
(2) distance
(3) local clustering
(4) multiple routes
(5) star spreaders
Network Diffusion & Peer Influence
Network diffusion features
• Much of the work on “core groups” or “at risk” populations focus on high-degree nodes. The
assumption is that high-degree nodes are likely to contact lots of people.
We need:
(1) reachability
(2) distance
(3) local clustering
(4) multiple routes
(5) star spreaders
Network Diffusion & Peer Influence
Network diffusion features
Network Diffusion & Peer Influence
Network diffusion features
Assortative mixing:
A more traditional way to think about “star” effects.
• Simulation study: How do these different features compare over a collection of observed nets?
• For each network trial:
• Fix dyadic transmission probability
• Randomly select a node as seed
• Trace the diffusion path across the network
• Measure speed & extent of spread
• Model extent of spread by structural characteristics
• First run: Add Health:
• simple diffusion process
• dyadic probability set to 0.08
• 500 trials in each network.
• Then expand to :
• Different assumptions of dyadic transmission probability
• Different data set (Facebook)
• Complex diffusion models
Network Diffusion & Peer Influence
Network diffusion features: simulation test
Network Diffusion & Peer Influence
Network diffusion features: simulation test
Network Diffusion & Peer Influence
Network diffusion features: simulation test
Define as a general measure of the “diffusion susceptibility” of a graph’s structure as the
ratio of the area under the observed curve to the area under the curve for a matching
random network. As this gets smaller than 1.0, you get effectively slower median
transmission.
Network Diffusion & Peer Influence
Network diffusion features: simulation test
Table 2. OLS Regression of Relative Diffusion Ratio on Network Structure
Variable Model 1 Model 2 Model 3 Model 4 Model 5
Intercept 1.62***
1.90***
1.02***
1.81***
1.71***
Connectivity
Distance -0.207***
-0.179***
-0.171***
Independent Paths -0.077***
-0.056***
-0.052***
Distance x Paths 0.023***
0.015***
0.016***
Clustering
Clustering Coefficient -0.692***
-0.653***
-0.454***
Grade Homophily -0.026**
-0.007 -0.009*
Peer Group Strength -0.868***
-0.141 -0.146
Degree Distribution
Degree Skew -0.023 -0.007 -0.002
Assortative Mixing -0.189*
-0.059 -0.071
Control Variables
Network Size/100 0.005***
-0.005***
-.005***
0.004*
0.002**
Proportion Isolated -0.007 -1.106***
-.984***
-0.300*
0.058
Non-Complete -0.006 -0.052*
-.078**
-0.006 0.018
Adj- R2
0.85 0.76 0.60 0.90 0.93
N 124 124 124 124 121
Network Diffusion & Peer Influence
Network diffusion features: simulation test
Table 2. OLS Regression of Relative Diffusion Ratio on Network Structure
Variable Model 1 Model 2 Model 3 Model 4 Model 5
Intercept 1.62***
1.90***
1.02***
1.81***
1.71***
Connectivity
Distance -0.207***
-0.179***
-0.171***
Independent Paths -0.077***
-0.056***
-0.052***
Distance x Paths 0.023***
0.015***
0.016***
Clustering
Clustering Coefficient -0.692***
-0.653***
-0.454***
Grade Homophily -0.026**
-0.007 -0.009*
Peer Group Strength -0.868***
-0.141 -0.146
Degree Distribution
Degree Skew -0.023 -0.007 -0.002
Assortative Mixing -0.189*
-0.059 -0.071
Control Variables
Network Size/100 0.005***
-0.005***
-.005***
0.004*
0.002**
Proportion Isolated -0.007 -1.106***
-.984***
-0.300*
0.058
Non-Complete -0.006 -0.052*
-.078**
-0.006 0.018
Adj- R2
0.85 0.76 0.60 0.90 0.93
N 124 124 124 124 121
Network Diffusion & Peer Influence
Network diffusion features: simulation test
Table 2. OLS Regression of Relative Diffusion Ratio on Network Structure
Variable Model 1 Model 2 Model 3 Model 4 Model 5
Intercept 1.62***
1.90***
1.02***
1.81***
1.71***
Connectivity
Distance -0.207***
-0.179***
-0.171***
Independent Paths -0.077***
-0.056***
-0.052***
Distance x Paths 0.023***
0.015***
0.016***
Clustering
Clustering Coefficient -0.692***
-0.653***
-0.454***
Grade Homophily -0.026**
-0.007 -0.009*
Peer Group Strength -0.868***
-0.141 -0.146
Degree Distribution
Degree Skew -0.023 -0.007 -0.002
Assortative Mixing -0.189*
-0.059 -0.071
Control Variables
Network Size/100 0.005***
-0.005***
-.005***
0.004*
0.002**
Proportion Isolated -0.007 -1.106***
-.984***
-0.300*
0.058
Non-Complete -0.006 -0.052*
-.078**
-0.006 0.018
Adj- R2
0.85 0.76 0.60 0.90 0.93
N 124 124 124 124 121
Network Diffusion & Peer Influence
Network diffusion features: simulation test
Figure 4. Relative Diffusion Ratio
By Distance and Number of Independent Paths
0.4
0.6
0.8
1
1.2
2.3 2.8 3.3 3.8 4.3 4.8 5.3 5.8 6.3
Average Path Length
Observed/Random
k=2
k=4
k=6
k=8
Figure 4. Relative Diffusion Ratio
By Distance and Number of Independent Paths
0.4
0.6
0.8
1
1.2
2.3 2.8 3.3 3.8 4.3 4.8 5.3 5.8 6.3
Average Path Length
Observed/Random
k=2
k=4
k=6
k=8
Network Diffusion & Peer Influence
Network diffusion features: simulation test
Traditional “core group” models have a local-vision understanding of risk: those
with lots of ties (high degree) are the focus for intervention and actions.
•In the short time-windows necessary for STD transfer, low-degree networks are
the relevant features for transmission. What sorts of networks emerge when
average degree (in the short run) is held to small numbers?
•How does the shape of the degree distribution matter? If activity is homogeneous
do we get fundamentally different networks than if it is very heterogeneous?
Network Diffusion & Peer Influence
A closer look at emerging connectivity
Partner
Distribution
Component
Size/Shape
Emergent Connectivity in low-degree networks
Network Diffusion & Peer Influence
A closer look at emerging connectivity
Network Diffusion & Peer Influence
A closer look at emerging connectivity
In both distributions, a giant
component & reconnected core
emerges as density increases, but
at very different speeds and
ultimate extent.
Network Diffusion & Peer Influence
A closer look at emerging connectivity
What distinguishes
these two distributions?
Shape
Network Diffusion & Peer Influence
A closer look at emerging connectivity
What distinguishes these two distributions?
Shape:
The scale-free network’s signature is the long-tail
So what effect does changes in the shape have on connectiv
Network Diffusion & Peer Influence
A closer look at emerging connectivity
 Volume 
DispersionxSkewness
Network Diffusion & Peer Influence
A closer look at emerging connectivity
Search Procedure:
1) Identify all valid degree distributions with
the given mean degree and a maximum of
6 w. brute force search.
2) Map them to this space
3) Simulate networks each degree distribution
4) Measure size of components &
Bicomponents
Network Diffusion & Peer Influence
A closer look at emerging connectivity
Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI),
Network Diffusion & Peer Influence
A closer look at emerging connectivity
Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI),
Network Diffusion & Peer Influence
A closer look at emerging connectivity
C:45%, B: 8.5%
Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI),
Network Diffusion & Peer Influence
A closer look at emerging connectivity
C:45%, B: 8.5%
Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI),
Network Diffusion & Peer Influence
A closer look at emerging connectivity
C:83%, B: 36%
Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI),
Network Diffusion & Peer Influence
A closer look at emerging connectivity
Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI),
C:99%, B: 86%
Network Diffusion & Peer Influence
A closer look at emerging connectivity
Largest Component
(at least 1 path)
Largest Bicomponent
(at least 2 paths)
Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI),
Network Diffusion & Peer Influence
A closer look at emerging connectivity
In addition to* the dyadic probability that one actor passes something to another
(pij), two factors affect flow through a network:
Topology
- the shape, or form, of the network
- Example: one actor cannot pass information to another unless they are
either directly or indirectly connected
Time
- the timing of contact matters
- Example: an actor cannot pass information he has not receive yet
*This is a big conditional! – lots of work on how the dyadic transmission rate may differ across
populations.
Key Question: What features of a network contribute most to diffusion potential?
Network Diffusion & Peer Influence
Relational Dynamics
Use simulation tools to explore the relative effects of structural
connectivity features
Three relevant networks
Discussions of network effects on STD spread often speak loosely of “the network.”
There are three relevant networks that are often conflated:
1) The contact network. The set of pairs of people connected by sexual contact. G(V,E).
2) The exposure network. A subset of the edges in the contact network where timing
makes it possible for one person to pass infection to another.
3) The transmission network. The subset of the exposure network where disease is actually
passed. In most cases this is a tree layered on (2) and rooted on a source/seed node.
Network Diffusion & Peer Influence
Relational Dynamics
Contact network: Everyone, it is a connected component
Who can “A” reach?
Network Diffusion & Peer Influence
Relational Dynamics
Discussions of network effects on STD spread often speak loosely of “the network.”
There are three relevant networks that are often conflated:
Three relevant networks
Exposure network: here, node “A” could reach up to 8 others
Who can “A” reach?
Network Diffusion & Peer Influence
Relational Dynamics
Discussions of network effects on STD spread often speak loosely of “the network.”
There are three relevant networks that are often conflated:
Three relevant networks
Transmission network: upper limit is 8 through the exposure links
(dark blue). Transmission is path dependent: if no transmission to
B, then also none to {K,L,O,J,M}
Who can “A” reach?
Exposable Link (from A’s p.o.v.)
Contact
Network Diffusion & Peer Influence
Relational Dynamics
Discussions of network effects on STD spread often speak loosely of “the network.”
There are three relevant networks that are often conflated:
Three relevant networks
The mapping between the contact network and the exposure network is based on
relational timing. In a dynamic network, edge timing determines if something can flow
down a path because things can only be passed forward in time.
Definitions:
Two edges are adjacent if they share a node.
A path is a sequence of adjacent edges (E1, E2, …Ed).
A time-ordered path is a sequence of adjacent edges where, for each pair of
edges in the sequence, the start time Si is less than or equal to Ej S(E1) <
E(E2)
Adjacent edges are concurrent if they share a node and have start and end
dates that overlap. This occurs if:
S(E2) < E(E1)
Concurrency
Network Diffusion & Peer Influence
Relational Dynamics
A B C
D
time 1 2 3 4 5 6 7 8 9 10
AB
BC
CE
E
CD
2 - 71 - 3
S(ab) E(ab)
S(bc)
E(bc)
S(ce) E(ce)
The mapping between the contact network and the exposure network is based on
relational timing. In a dynamic network, edge timing determines if something can flow
down a path because things can only be passed forward in time.
Concurrency
Network Diffusion & Peer Influence
Relational Dynamics
The constraints of time-ordered paths change our understanding of the system structure
of the network. Paths make a network a system: linking actors together through indirect
connections. Relational timing changes how paths cumulate in networks.
Indirect connectivity is no longer transitive:
A B C D1 - 2 3 - 4 1 - 2
Here A can reach C, and C and reach D. But A cannot reach D (nor D A). Why?
Because any infection A passes to C would have happened after the relation between C
and D ended.
A B C D1 - 2 3 - 4 1 - 2
Network Diffusion & Peer Influence
Relational Dynamics
Edge time structures are characterized by sequence, duration and overlap.
Paths between i and j, have length and duration, but these need not be
symmetric even if the constituent edges are symmetric.
Network Diffusion & Peer Influence
Relational Dynamics
1 2 2 2 2 2 1
1 1 2 2 2 2 2
2 1 1 2 2 2 2
2 2 1 1 2 2 2
2 2 2 1 1 2 2
2 2 2 2 1 1 2
2 2 2 2 2 1 1
1 2 2 2 2 2 1
Implied Contact Network of 8 people in a ring
All relations Concurrent
Reachability = 1.0
Network Diffusion & Peer Influence
Relational Dynamics
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1
1 1 1 1
1 1 1
1 1
Implied Contact Network of 8 people in a ring
Serial Monogamy (1)
1
2
3
7
6
5
8
4
Reachability = 0.71
Network Diffusion & Peer Influence
Relational Dynamics
1 1 1
1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1
1 1 1
1 1 1 1 1
1 1 1 1 1
Implied Contact Network of 8 people in a ring
Mixed Concurrent
2
2
1
1
2
2
3
3
Reachability = 0.57
Network Diffusion & Peer Influence
Relational Dynamics
Implied Contact Network of 8 people in a ring
Serial Monogamy (3)
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1
2
1
1
2
1
2
2
Reachability = 0.43
Network Diffusion & Peer Influence
Relational Dynamics
1
2
1
1
2
1
2
2
Timing alone can change mean reachability from
1.0 when all ties are concurrent to 0.42.
In general, ignoring time order is equivalent to
assuming all relations occur simultaneously –
assumes perfect concurrency across all relations.
Network Diffusion & Peer Influence
Relational Dynamics
A B C D E F
A 0 1 2 2 4 1
B 1 0 1 2 3 2
C 0 1 0 1 2 2
D 0 0 1 0 1 1
E 0 0 0 1 0 2
F 1 0 0 1 0 0
While a is 2 steps from d, and d is 1 step from e, a and e
are 4 steps apart.
This is because the shorter path from a to e emerges after
the path from d to e ended.
4
2
1
Path distances no longer simply add
Network Diffusion & Peer Influence
Relational Dynamics
A
B C
D
3 - 4
E
The geodesic from A to D is AE, ED and is two steps long.
But the fastest path would be AB, BC, CD, which while 3 steps long
could get there by time 5 compared to time 7.
I ignore this feature for the remainder…
Aside: “shortest” can now be replaced with “fastest”
Network Diffusion & Peer Influence
Relational Dynamics
Concurrency affects exposure by making paths symmetric, which increases exposure down
multiple “branches” of a contact sequence.
Consider a simplified example:
All edges concurrent
a b c d e f g h i j k l m
a . 1 1 1 1 1 1 1 1 1 1 1 1
b 1 . 1 1 1 1 1 1 1 1 1 1 1
c 1 1 . 1 1 1 1 1 1 1 1 1 1
d 1 1 1 . 1 1 1 1 1 1 1 1 1
e 1 1 1 1 . 1 1 1 1 1 1 1 1
f 1 1 1 1 1 . 1 1 1 1 1 1 1
g 1 1 1 1 1 1 . 1 1 1 1 1 1
h 1 1 1 1 1 1 1 . 1 1 1 1 1
i 1 1 1 1 1 1 1 1 . 1 1 1 1
j 1 1 1 1 1 1 1 1 1 . 1 1 1
k 1 1 1 1 1 1 1 1 1 1 . 1 1
l 1 1 1 1 1 1 1 1 1 1 1 . 1
m 1 1 1 1 1 1 1 1 1 1 1 1 .
Network Diffusion & Peer Influence
Relational Dynamics
All edges except gd concurrent
a b c d e f g h i j k l m
a . 1 1 1 1 1 1 1 1 1 1 1 1
b 1 . 1 1 1 1 1 1 1 1 1 1 1
c 1 1 . 1 1 1 1 1 1 1 1 1 1
d 1 1 1 . 1 1 1 1 1 1 1 1 1
e 1 1 1 1 . 1 1 1 1 1 1 1 1
f 1 1 1 1 1 . 1 1 1 1 1 1 1
g 0 0 0 1 0 0 . 1 1 1 1 1 1
h 0 0 0 1 0 0 1 . 1 1 1 1 1
i 0 0 0 1 0 0 1 1 . 1 1 1 1
j 0 0 0 1 0 0 1 1 1 . 1 1 1
k 0 0 0 1 0 0 1 1 1 1 . 1 1
l 0 0 0 1 0 0 1 1 1 1 1 . 1
m 0 0 0 1 0 0 1 1 1 1 1 1 .
Concurrency affects exposure by making paths symmetric, which increases exposure down
multiple “branches” of a contact sequence.
Consider a simplified example:
Network Diffusion & Peer Influence
Relational Dynamics
Concurrency affects exposure by making paths symmetric, which increases
exposure down multiple “branches” of a contact sequence.
Consider a simplified example:
All edges gd after all others:
a b c d e f g h i j k l m
a . 1 1 1 1 1 1 0 0 0 0 0 0
b 1 . 1 1 1 1 1 0 0 0 0 0 0
c 1 1 . 1 1 1 1 0 0 0 0 0 0
d 1 1 1 . 1 1 1 0 0 0 0 0 0
e 1 1 1 1 . 1 1 0 0 0 0 0 0
f 1 1 1 1 1 . 1 0 0 0 0 0 0
g 0 0 0 1 0 0 . 1 1 1 1 1 1
h 0 0 0 0 0 0 1 . 1 1 1 1 1
i 0 0 0 0 0 0 1 1 . 1 1 1 1
j 0 0 0 0 0 0 1 1 1 . 1 1 1
k 0 0 0 0 0 0 1 1 1 1 . 1 1
l 0 0 0 0 0 0 1 1 1 1 1 . 1
m 0 0 0 0 0 0 1 1 1 1 1 1 .
The concurrency status of {dg}
determines which “side” of the graph
is exposed. Note this effect happens
at the system level – the correlation
between exposure and node-level
timing is essentially zero (d/g
excepted)
Network Diffusion & Peer Influence
Relational Dynamics
Resulting infection trace from a simulation (Morris et al, AJPH 2010).
Observed infection paths from 10 seeds in
an STD simulation, edges coded for
concurrency status.
Network Diffusion & Peer Influence
Relational Dynamics
Resulting infection trace from a simulation (Morris et al, AJPH 2010).
Network Diffusion & Peer Influence
Relational Dynamics
Observed infection paths from 10 seeds in
an STD simulation, edges coded for
concurrency status.
Timing constrains potential diffusion paths in networks, since bits can flow through edges that have
ended.
This means that:
• Structural paths are not equivalent to the diffusion-relevant path set.
• Network distances don’t build on each other.
• Weakly connected components overlap without diffusion reaching across sets.
• Small changes in edge timing can have dramatic effects on overall diffusion
• Diffusion potential is maximized when edges are concurrent and minimized when they are
“inter-woven” to limit reachability.
Combined, this means that many of our standard path-based network measures will be incorrect on
dynamic graphs.
Network Diffusion & Peer Influence
Relational Dynamics
The distribution of paths is important for many of the measures we typically construct on
networks, and these will be change if timing is taken into consideration:
Centrality:
Closeness centrality
Path Centrality
Information Centrality
Betweenness centrality
Network Topography
Clustering
Path Distance
Groups & Roles:
Correspondence between degree-based position and reach-based position
Structural Cohesion & Embeddedness
Opportunities for Time-based block-models (similar reachability profiles)
In general, any measures that take the systems nature of the graph into account will differ.
Structural measurement implications
Network Diffusion & Peer Influence
Relational Dynamics – other implication of dynamic nets (briefly)
New versions of classic reachability measures:
1) Temporal reach: The ij cell = 1 if i can reach j through time.
2) Temporal geodesic: The ij cell equals the number of steps in the shortest path linking i to j
over time.
3) Temporal paths: The ij cell equals the number of time-ordered paths linking i to j.
These will only equal the standard versions when all ties are concurrent.
Duration explicit measures
4) Quickest path: The ij cell equals the shortest time within which i could reach j.
5) Earliest path: The ij cell equals the real-clock time when i could first reach j.
6) Latest path: The ij cell equals the real-clock time when i could last reach j.
7) Exposure duration: The ij cell equals the longest (shortest) interval of time over which i
could transfer a good to j.
Each of these also imply different types of “betweenness” roles for nodes or edges, such as a
“limiting time” edge, which would be the edge whose comparatively short duration places
the greatest limits on other paths.
Structural measurement implications
Network Diffusion & Peer Influence
Relational Dynamics – other implication of dynamic nets (briefly)
Topology & Time interact: How relational sequencing affects diffusion is
conditioned by the structural patterns of relations.
Examples:
- Time limitations mean star nodes can’t interact with everyone at each
time tick; effects of high degree are thus limited by schedule/availabiltiy
- If within-cluster ties are also more frequent than between cluster ties,
then the effects of communities will be magnified.
-Multiple connectivity should provide routes around breaks built by
temporal sequence.
Network Diffusion & Peer Influence
Structural Moderators of Timing Effects
Measures
Dependent variable: Reachability in the exposure graph. This is the proportion of
pairs in the network that are reachable in time.
Exposure Graph Density
Network Diffusion & Peer Influence
Structural Moderators of Timing Effects
Measures: Independent variables
Features of the topology. Of key interest is the level of structural cohesion.
1 2 3 4 5 6 7 8 9
1 -- 1 1 1 1 1 1 1 1
2 1 -- 2 2 2 1 1 1 1
3 1 2 -- 2 2 1 1 1 1
4 1 2 2 -- 2 1 1 1 1
5 1 2 2 2 -- 1 1 1 1
6 1 1 1 1 1 -- 2 2 1
7 1 1 1 1 1 2 -- 2 1
8 1 1 1 1 1 2 2 -- 1
9 1 1 1 1 1 1 1 1 --
Cell Value = highest k-connected
component pair belongs to.
Average = 1.25
Network Diffusion & Peer Influence
Structural Moderators of Timing Effects
1 2 3 4 5 6 7 8 9
1 -- 1 1 1 1 1 1 1 1
2 1 -- 3 3 3 2 2 2 1
3 1 3 -- 3 3 2 2 2 1
4 1 3 3 -- 3 2 2 2 1
5 1 3 3 3 -- 2 2 2 1
6 1 2 2 2 2 -- 2 2 1
7 1 2 2 2 2 2 -- 2 1
8 1 2 2 2 2 2 2 -- 1
9 1 1 1 1 1 1 1 1 --
Cell Value = highest k-connected
component pair belongs to.
Average = 1.6
Measures: Independent variables
Features of the topology. Of key interest is the level of structural cohesion.
Network Diffusion & Peer Influence
Structural Moderators of Timing Effects
Measures: Independent variables
Features of the topology. Of key interest is the level of structural cohesion.
Average connectivity
Network Diffusion & Peer Influence
Structural Moderators of Timing Effects
Volume Distance Connectivity
Nodes: 148
Mean Deg: 6.16
Density: 0.042
Centralization: 0.187
Nodes: 80
Mean Deg: 5.27
Density: 0.067
Centralization: 0.373
Nodes: 154
Mean Deg: 3.71
Density: 0.025
Centralization: 0.147
Nodes: 128
Mean Deg: 3.39
Density: 0.027
Centralization: 0.205
Mean: 3.59
Diameter: 5
Centralization: 0.312
Mean: 3.02
Diameter: 5
Centralization: 0.413
Mean: 4.99
Diameter: 8
Centralization: 0.259
Mean: 4.55
Diameter: 6
Centralization: 0.301
Largest BC: 0.51
Pairwise K: 1.57
Largest BC: 0.33
Pairwise K: 1.34
Largest BC: 0.08
Pairwise K: 1.07
Largest BC:
Pairwise K: 1.06
Exemplar independent variables
“HighCohesive”“LowCohesive”Network Diffusion & Peer Influence
Structural Moderators of Timing Effects
Network Diffusion & Peer Influence
Structural Moderators of Timing Effects
“Low Cohesive”
Network Diffusion & Peer Influence
Structural Moderators of Timing Effects
Proportion of relations concurrent
DensityoftheExposureNetwork
Network Diffusion & Peer Influence
Structural Moderators of Timing Effects
Colors=different
nets
Panels=cohesion
level
Proportion of relations concurrent
DensityoftheExposureNetwork
Network Diffusion & Peer Influence
Structural Moderators of Timing Effects
Network Diffusion & Peer Influence
Structural Moderators of Timing Effects
1. Concurrency has a necessarily positive effect on potential diffusion
exposure
1. This implies that we should see greater transmission given greater
concurrency
2. This works by creating “multiple routes” in the exposure path
structure
2. Structural cohesion captures multiple routes in the contact graph
1. Higher levels of cohesion increase exposure by directly increasing
the underlying transmission substrate
3. There is a negative interaction between cohesion and concurrency: as
cohesion increases, the relative returns to concurrency decrease.
1. But this comes at the cost of a higher base-level of exposure.
Network Diffusion & Peer Influence
Structural Moderators of Timing Effects
Network Diffusion & Peer Influence
Structural Transmission Dynamics: beyond disease diffusion
Complex Contagion
Thus far we have focused on a “simple” dyadic diffusion parameter, pij, where the
probability of passing/receiving the bit is purely dependent on discordant status of
the dyad, sometimes called the “independent cascade model” (), which suggests a
monotonic relation between the number of times you are exposed through peers.
High exposure could be due to repeated interaction with one person or weak
interaction with many, effectively equating:
Alternative models exist. Under “complex contagion” for example, the likelihood
that I accept the bit that flows through the network depends on the proportion of my
peers that have the bit.
Network Diffusion & Peer Influence
Structural Transmission Dynamics: beyond disease diffusion
1
1
2
3
Complex Contagion
Assume adoption requires k neighbors having adopted, then transmission can only
occur within dense clusters:
Network Diffusion & Peer Influence
Structural Transmission Dynamics: beyond disease diffusion
Complex Contagion
Assume adoption requires k neighbors having adopted, then transmission can only
occur within dense clusters:
Assume pij=1, k=2, starting nodes in yellow
Network Diffusion & Peer Influence
Structural Transmission Dynamics: beyond disease diffusion
Complex Contagion
Assume adoption requires k neighbors having adopted, then transmission can only
occur within dense clusters:
For this network under weak complex diffusion (k=2), the maximum risk size is 8.
Network Diffusion & Peer Influence
Structural Transmission Dynamics: beyond disease diffusion
Complex Contagion
Assume adoption requires k neighbors having adopted, then transmission can only
occur within dense clusters:
For this network under weak complex diffusion (k=2), the maximum risk size is reaches 98%.
One of the Prosper
schools:
Start
Network Diffusion & Peer Influence
Structural Transmission Dynamics: beyond disease diffusion
Complex Contagion
Can lead to widely varying sizes of potential diffusion cascades. Here’s the
distribution across all PROPSPER schools:
Distribution is largely bimodal (even with a connected pair start)
Network Diffusion & Peer Influence
Structural Transmission Dynamics: beyond disease diffusion
Complex Contagion
Can lead to widely varying sizes of potential diffusion cascades. Here’s the
distribution across all PROPSPER schools:
The governing factors are (a) curved effect of local redundancy and (b) structural cohesion
Network Average Proportion Reached
k=2 complex contagion
MeanCascadeSize
Coh=0.3
Coh=1.2
Coh=2.2
Coh=3.2
Coh=4.1
Network Diffusion & Peer Influence
Structural Transmission Dynamics: beyond disease diffusion
Complex Contagion
Does get used for real health
work:
Here , authors assume a CC
process, seeded with observed
depressive cases, turn that into
a Markov model and ask what
parameters would maximize fit
from simulated to observed.
Network Diffusion & Peer Influence
Structural Transmission Dynamics: beyond disease diffusion
Complex diffusion is just the most well studied of the options that combine
transmission with some pairwise positional feature. This is a wide-open
area for future research.
The basic idea is that transmission is increased/decreased if there is some
third structural property that the susceptible & infected pair share.
This leads us into the general problem of peer influence models…when do
peers change each other’s behaviors?
Background:
• Long standing research interest in how our relations shape our attitudes
and behaviors.
• Most often assumed mechanism is that people (through conversation or
similar) change each others beliefs/opinions, which changes behavior.
This implies that position in a communication network should be related
to attitudes.
• Alternatives:
• Modeling behavior: ego copies behavior of alter to gain respect,
esteem, etc.
• Distinction: Ego tries to be different from (some) alter to gain
respect, esteem, etc.
• Access: Ego wants to do Y, but can only do so because alter provides
access (say, being old enough to buy cigarettes).
Network Diffusion & Peer Influence
Peer Influence Dynamics
Background:
• Early work was ego-centric – people informed on their peers
•Seems to have inflated PI effects by ~50% or so…either through projection of
ego behavior onto peers or selective interaction (what alters do with ego may be
different than what alter does all the time).
•Then to cross sectional associations based on alter self-reports
•Better, but still likely conflates selection with influence
•Next to dynamic models:
•Ego Behavior(t) ~ f(ego behavior(t-1) + alter behavior (t-1) + controls
•Much better; still debate on (a) correct estimation functions, (b) unobserved
selection features that confound causal inference.
•Development of Actor-oriented models (SIENA)
Network Diffusion & Peer Influence
Peer Influence Dynamics
Background:
•Finally: Experimental manipulation of peer exposure
•“Gold standard” for isolation of peer effects
•Likely strongly underestimates effects (as measure intent to treat, not take-
up of treatment, since people may not care about relations that can be
manipulated).
b(Peer(y)): Ego Inform < Alter Inform < Cross Sectional < Dynamic < Experimental. Still often
find peer effects, but my sense is that we’ve (strongly) over-corrected at this point.
Network Diffusion & Peer Influence
Peer Influence Dynamics
Freidkin’s Structural Theory of Social Influence :
Two-part model:
Beliefs are a function of two sources:
a) Individual characteristics
•Gender, Age, Race, Education, Etc. Standard sociology
b) Interpersonal influences
•Actors negotiate with others
Network Diffusion & Peer Influence
Peer Influence Dynamics
XBY )1( (1)
)1()1()(
)1( YWYY αα Tt
 
(2)
Y(1) = an N x M matrix of initial opinions on M issues for N
actors
X = an N x K matrix of K exogenous variable that affect Y
B = a K x M matrix of coefficients relating X to Y
a = a weight of the strength of endogenous interpersonal
influences
W = an N x N matrix of interpersonal influences
Network Diffusion & Peer Influence
Peer Influence Dynamics
XBY )1( (1)
This is the standard sociology model for explaining anything: the General Linear Model.
It says that a dependent variable (Y) is some function (B) of a set of independent
variables (X). At the individual level, the model says that:

k
kiki BXY
Usually, one of the X variables is e, the model error term.
Network Diffusion & Peer Influence
Peer Influence Dynamics
)1()1()(
)1( YWYY αα Tt
 
(2)
This part of the model taps social influence. It says that each person’s final opinion is
a weighted average of their own initial opinions
)1(
)1( Yα
And the opinions of those they communicate with (which can include their own current
opinions)
)1( T
αWY
Network Diffusion & Peer Influence
Peer Influence Dynamics
The key to the peer influence part of the model is W, a matrix of
interpersonal weights. W is a function of the communication structure of the
network, and is usually a transformation of the adjacency matrix. In general:
 

j
ij
ij
w
w
1
10
Various specifications of the model change the value of wii, the extent to which
one weighs their own current opinion and the relative weight of alters.
Network Diffusion & Peer Influence
Peer Influence Dynamics
1 2
3
4
1 2 3 4
1 1 1 1 0
2 1 1 1 0
3 1 1 1 1
4 0 0 1 1
1 2 3 4
1 .33 .33 .33 0
2 .33 .33 .33 0
3 .25 .25 .25 .25
4 0 0 .50 .50
1 2 3 4
1 .50 .25 .25 0
2 .25 .50 .25 0
3 .20 .20 .40 .20
4 0 0 .33 .67
Even
2*self
1 2 3 4
1 .50 .25 .25 0
2 .25 .50 .25 0
3 .17 .17 .50 .17
4 0 0 .50 .50
degree
Self weight:
1 2 3 4
1 2 1 1 0
2 1 2 1 0
3 1 1 2 1
4 0 0 1 2
1 2 3 4
1 2 1 1 0
2 1 2 1 0
3 1 1 3 1
4 0 0 1 1
Network Diffusion & Peer Influence
Peer Influence Dynamics
)1()1()(
)1( YWYY αα Tt
 
Formal Properties of the model
When interpersonal influence is complete, model reduces to:
)1(
)1()1()(
01




T
Tt
WY
YWYY
When interpersonal influence is absent, model reduces to:
)1(
)1()1()(
0
Y
YWYY

 Tt
(2)
Network Diffusion & Peer Influence
Peer Influence Dynamics
Formal Properties of the model
The model is directly related to spatial econometric models:
If we allow the model to run over t and W remains constant:
XBWYY )1()()(
αα  
eb  
XWYY
~)()(
α
Where the two coefficients (a and b) are estimated directly (See Doreian,
1982, SMR).
This is the linear network auto correlation model, best bet with cross-sectional
data (and randomization trick to estimate se)
Network Diffusion & Peer Influence
Peer Influence Dynamics
Simple example
1 2
3
4
1 2 3 4
1 .33 .33 .33 0
2 .33 .33 .33 0
3 .25 .25 .25 .25
4 0 0 .50 .50
Y
1
3
5
7
a = .8
T: 0 1 2 3 4 5 6 7
1.00 2.60 2.81 2.93 2.98 3.00 3.01 3.01
3.00 3.00 3.21 3.33 3.38 3.40 3.41 3.41
5.00 4.20 4.20 4.16 4.14 4.14 4.13 4.13
7.00 6.20 5.56 5.30 5.18 5.13 5.11 5.10
By t=7, still variability in Y
Network Diffusion & Peer Influence
Peer Influence Dynamics
1 2
3
4
1 2 3 4
1 .33 .33 .33 0
2 .33 .33 .33 0
3 .25 .25 .25 .25
4 0 0 .50 .50
Y
1
3
5
7
a = 1.0
1.00 3.00 3.33 3.56 3.68 3.74 3.78 3.81
3.00 3.00 3.33 3.56 3.68 3.74 3.78 3.81
5.00 4.00 4.00 3.92 3.88 3.86 3.85 3.84
7.00 6.00 5.00 4.50 4.21 4.05 3.95 3.90
By t=7, almost no variability in Y
T: 0 1 2 3 4 5 6 7
Simple example
Network Diffusion & Peer Influence
Peer Influence Dynamics
Extended example: building intuition
Consider a network with three cohesive groups, and an initially random distribution of
opinions:
Network Diffusion & Peer Influence
Peer Influence Dynamics
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Extended example: building intuition
Consider a network with three cohesive groups, and an initially random distribution of opinions:
Now weight in-group ties higher than between group ties
Network Diffusion & Peer Influence
Peer Influence Dynamics
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations, in-group tie: 2
Consider the implications for populations of different structures. For example, we might
have two groups, a large orthodox population and a small heterodox population. We can
imagine the groups mixing in various levels:
Little Mixing Moderate Mixing Heavy Mixing
.95 .05
.05 .02
.95 .008
.008 .02
.95 .001
.001 .02
Heterodox: 10 people
Orthodox: 100 People
Network Diffusion & Peer Influence
Peer Influence Dynamics
Light Heavy
Moderate
Light mixing
Light mixing
Light mixing
Light mixing
Light mixing
Light mixing
Moderate mixing
Moderate mixing
Moderate mixing
Moderate mixing
Moderate mixing
Moderate mixing
High mixing
High mixing
High mixing
High mixing
High mixing
High mixing
In an unbalanced situation (small group vs large group) the extent of contact
can easily overwhelm the small group. Applications of this idea are evident
in:
•Missionary work (Must be certain to send missionaries out into the
world with strong in-group contacts)
•Overcoming deviant culture (I.e. youth gangs vs. adults)
•This is also the mechanism behind why most youth peer influence
is a *good* thing – most youth are well behavior and civic
minded…so are exerting positive influences on their peers.
Network Diffusion & Peer Influence
Peer Influence Dynamics
Friedkin (1998) generalizes the model so that alpha varies across people.
(1) simply changing a to a vector (A), which then changes each person’s
opinion directly
(2) by linking the self weight (wii) to alpha.
)1()1()(
)( YAIAWYY  Tt
Were A is a diagonal matrix of endogenous weights, with 0 < aii < 1. A further
restriction on the model sets wii = 1-aii
This leads to a great deal more flexibility in the theory, and some interesting
insights. Consider the case of group opinion leaders with unchanging opinions
(I.e. many people have high aii, while a few have low):
Network Diffusion & Peer Influence
Peer Influence Dynamics
Group 1
Leaders
Group 2
Leaders
Group 3
Leaders
Peer Opinion Leaders
Peer Opinion Leaders
Peer Opinion Leaders
Peer Opinion Leaders
Peer Opinion Leaders
Peer Opinion Leaders
Further extensions of the model might:
• Time dependent a: people likely value other’s opinions more early than later in a
decision context
• Interact a with XB: people’s self weights are a function of their behaviors &
attributes
• Make W dependent on structure of the network (weight transitive ties greater
than intransitive ties, for example)
• Time dependent W: The network of contacts does not remain constant, but is
dynamic, meaning that influence likely moves unevenly through the network
• And others likely abound….
Network Diffusion & Peer Influence
Peer Influence Dynamics
There are two common ways to test for peer associations through networks.
The first estimates the parameters (a and b) of the network autocorrelation model directly,
the second transforms the network into a dyadic model, predicting similarity among
actors.
eb  
XWYY
~)()(
α
Peer influence model:
Network Diffusion & Peer Influence
Peer Influence Dynamics
This is the linear network autocorrelation model, and as specified, the
model makes strong assumptions about equilibrium opinion and static
relations.
 Some variants on this also expand e to include alternative
autocorrelation in the error structure.
There are two common ways to test for peer associations through networks.
The first estimates the parameters (a and b) of the network autocorrelation model directly,
the second transforms the network into a dyadic model, predicting similarity among
actors.
eb  
XWYY
~)()(
α
Peer influence model:
Network Diffusion & Peer Influence
Peer Influence Dynamics
Note that since WY is a a simple vector -- weighted mean of friends Y -- which can be
constructed and added to your GLM model. That is, multiple Y by a W matrix, and run the
regression with WY as a new variable, and the regression coefficient is an estimate of a. This is
what Doriean calls the QAD estimate of peer influence.
It’s wrong, a will be biased, but it’s often not terribly wrong if most obvious selection factors are
built int0 X
An obvious problem with this specification is that cases are, by definition, not
independent, hence “network autocorrelation” terminology.
In practice, the QAD approach (perhaps combined with a GLS estimator)
results in empirical estimates that are “virtually indistinguishable” from MLE
(Doreian et al, 1984)
The proper way to estimate the peer equation is to use maximum likelihood
estimates, and Doreian gives the formulas for this in his paper, and Carter Butts
has implemented in in R with the LNAM procedure.
An alternative is to use non-parametric approaches, such as the Quadratic
Assignment Procedure, to estimate the effects.
Network Diffusion & Peer Influence
Peer Influence Dynamics
Peer influence through Dyad Models
Another way to get at peer influence is not through the level of Y, but by assessing the
similarity of connected peers. Recall the simulated example: peer influence is reflected in
how close points are to each other.
Network Diffusion & Peer Influence
Peer Influence Dynamics
Peer influence through Dyad Models
The model is now expressed at the dyad level as:
ij
k
kkijij eXbAbbY  10
Where Y is a matrix of similarities, A is an adjacency matrix, and Xk is a matrix of similarities
on attributes
Advantages include ease of specifying relation-specific similarity functions. You can add
different features of a relation by adjusting/adding “Aij” variables.
Disadvantage is that now in addition to network autocorrelation, you have repeated cases (on
both sides).
But these can be dealt with using non-parametric modeling & testing techniques (QAP, for
example). (which we will go over this afternoon)
Network Diffusion & Peer Influence
Peer Influence Dynamics
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Used the friend/relative
tracking data from a
larger heart-health study
to identify network
contacts, including
friends.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Used the friend/relative
tracking data from a
larger heart-health study
to identify network
contacts, including
friends.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Used the friend/relative
tracking data from a
larger heart-health study
to identify network
contacts, including
friends.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Used the friend/relative
tracking data from a
larger heart-health study
to identify network
contacts, including
friends.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Used the friend/relative
tracking data from a
larger heart-health study
to identify network
contacts, including
friends.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Used the friend/relative
tracking data from a
larger heart-health study
to identify network
contacts, including
friends.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Used the friend/relative
tracking data from a
larger heart-health study
to identify network
contacts, including
friends.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
The network shows significant evidence of weight-homophily
Used the friend/relative
tracking data from a
larger heart-health study
to identify network
contacts, including
friends.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Effects of peer obesity on ego, by peer type
Edge-wise regressions of the form:
ControlsEgoAltAltEgo previouspreviousCurrentCurrent  )()()( 321 bbb
Ego is repeated for all alters; models include random effects on ego id
Used the friend/relative
tracking data from a
larger heart-health study
to identify network
contacts, including
friends.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
This modeling strategy
pools observations on
edges and estimates a
global effect net of
change in ego/alter as a
control. Here color is a
single ego, number is
wave (only 2 egos and 3
waves represented).
Effects of peer obesity on ego, by peer type
ControlsEgoAltAltEgo previouspreviousCurrentCurrent  )()()( 321 bbb
1
Ego-Current
Alter Current
1 1
2 2 2
3 3 3
1 1 1
2 2 2
3 3 3
Peer Effect
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Alterative specifications
include using change-
change models and
allowing for a random
effect of peers. This
allows for greater
variability in peer effects,
and the potential to model
differences.
ee
currentpreviouseCurrentprevious ControlstAltAlEgoEgo
bb
b

 )()(
1
Ego-Current
Alter Current
1 1
2 2 2
3 3 3
1 1 1
2 2 2
3 3 3
b
be1
be1
Effects of peer obesity on ego, by peer type
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
ee
previouspreviousCurrenteCurrent ControlsEgoAltAltEgo
bb
bbb

 )()()( 32
Or difference
models:
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Critiques of C&F
The C&F studies – of obesity, but also other work on the FHS data – turn on the validity of the
causal association.
All turn on some issue of model miss-specification, typically:
• Can’t truly distinguish a network effect from other sources of common influence
• “Selection” (“homophily”) or “Common influence” (“Shared environment”)
• The most strident work in this area (Salizi
• Statistical errors
• Misinterpretation of confidence intervals
• Poorly specified/estimated models
C&H do a nice job of laying out their responses here:
http://jhfowler.ucsd.edu/examining_dynamic_social_networks.pdf and here:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2597062/
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Critiques of C&F
Cohen-Cole, E. and Fletcher, J. M. (2008). Detecting implausible social network effects
in acne, height, and headaches: longitudinal analysis. British Medical Journal 337
a2533.
Use the same models as C&F
on Add Health to show that
things which are theoretically
unlikely to be contagious
appear to be in this form of
model.
Note these coefficients are
substantially smaller than C&F
and only significant at the 0.1
level; and not robust to any
sensitivity analysis.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Critiques of C&F
Lyons, 2011.
1) C&F claim that differences in directional effects support a PI story:
• C& F: While mutual friends and
egoalter friends are > 0,
alterego is not, means ego is
emulating alter.
• Lyons notes these CIs
overlap too much to make
any claim about
distinguishing them from
each other.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Critiques of C&F Lyons, 2011.
2) Insufficient controls for Homophily
• C& F: Use of alter’s lagged Y to control for homophily. Logic is that any feature
that selected us to be friends at t-1 would have had it’s effect then.
• Lyons notes that current and lagged have opposite signs, which seems
suspect, and anyway is an insufficient control. He’s likely right here…
3) Directionality cannot distinguish the source of association
• C& F: the ordering: mutual, egoalter, alterego suggests an “esteem” model,
where ego copies the behavior of alter.
• Lyons argues that we would expect the same logic from a simple “foci” of
similarity. I don’t find this argument convincing.
4) Random permutation tests cannot establish 3-degree rule
• C& F: Association between alters at 1, 2, 3 degrees of separation are higher than
we’d expect by chance, based on a permutation test.
• Lyons invalid if the data are incomplete, which they certainly are. I don’t
find this argument convincing…data are always incomplete…
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Critiques of C&F Lyons, 2011.
5) The models are statistically inconsistent (if not incoherent)
• C& F: Use separate models for each type of tie, with random effects on ego.
• Lyons notes that these really should be treated as simultaneous equations,
with shared error structures and so forth. Doing so (a) leads to unidentified
models that must force the estimation of the peer effect to 0. That observed
^0 indicates something amiss.
• Strikes me as a bit down in the weeds and I’m not convinced here that he’s
critiquing them for what they are really doing (argues there are more
equations than data, which is patently not true).
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Critiques of C&F Lyons, 2011.
My sense is that the strategy C&F took was not fundamentally misguided, but the model
specification is probably thin; certainly in the obesity paper – less so in some of the later papers
appearing after these debates.
Ideally you’d have a much better direct model for selection – perhaps even a separate two-stage
model (see the Siena module), but here there are very limited observational controls, which would
have been easy to add. In later specifications, they do add fixed effects for ego and still find similar
results.
Commenting on the debate on SocNET – and a related conclusion that only experiments could
provide valid inference – Tom Snijders says:
“The logical consequence of this is that we are stuck with imperfect methods. Lyons argues as
though only perfect methods are acceptable, and while applauding such lofty ideals I still believe
that we should accept imperfection, in life as in science. Progress is made by discussion and
improvement of imperfections, not by their eradication.”
For a full general discussion, see : https://www.lists.ufl.edu/cgi-bin/wa?A2=ind1106&L=SOCNET&P=R11428
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Shalizi & Thomas: PI is *generally* confounded
So long as there is an unobserved X that causes both ties and behavior, the
effect of peers is unidentified.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Shalizi & Thomas: PI is *generally* confounded
Only route out is to make X fully informed (or informing) by an observable Z….but
realistically there are few things that (a) cause behavior exclusively without any
selection pressure (a) or cause ties exclusively without any influence pressure (b)
(though note b is what experimental assignments do)
(X causes Z, not Y directly) (X causes A, not Y directly)
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Shalizi & Thomas: PI is *generally* confounded
Should be noted that this is true for *any* effect – there’s always the
potential that an unobserved latent variable is creating a spurious effect;
This sort of work argues that the only solution is to use experimental (or,
sometimes, propensity score style models)…but that’s simply not always
feasible practically.
We need to beware of making the best the enemy of the good enough…lest
we make no progress at all…
Willard Van Quine, professor of philosophy and mathematics emeritus from
Harvard University who is regarded as one of the most famous philosophers
in the world, wrote his doctoral thesis on a 1927 Remington typewriter,
which he still uses. However, he "had an operation on it" to change a few
keys to accommodate special symbols. "I found I could do without the
second period, the second comma -- and the question mark.”
"You don't miss the question mark?”
"Well, you see, I deal in certainties."
Selection or Influence?
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Is it all selection
•What do we know about how friendships form?
•Opportunity / focal factors
- Being members of the same group
- In the same class
- On the same team
- Members of the same church
•Structural Relationship factors
- Reciprocity
- Social Balance
•Behavior Homophily
- Smoking
- Drinking
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Network Model Coefficients, In school Networks
Network Diffusion & Peer Influence
How to correct this problem?
•Essentially, this is an omitted variable problem, and my “solution” has
been to identify as many potentially relevant alternative variables as I can
find.
•The strongest possible correction is to use fixed-effects* models that
control for all non-varying individual covariates. These have their own
problems…
•Dual model for influence & selection.
•Two-stage model “Heckman” sorts of models
•Dynamic SAOM models
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
*“Adding fixed effects to dynamic panel models with many subjects and few repeat observations creates severe bias towards zero coefficients. This has
been demonstrated both analytically (Nickell 1981) and through simulations (Nerlove 1971) for OLS and other regression models and has been well-known
by social scientists, including economists, for a very long time. In fact, CCF even note that they do not add fixed effects to their logit regression model for
this reason, but they strangely assert that fixed effects are necessary in the OLS model.” Estimating Peer Effects on Health in Social Networks : A
Response to Cohen-Cole and Fletcher; Trogdon, Nonnemaker, Pais J.H. Fowler, PhD and N.A. Christakis, MD, PhD
• Causal status of such similarity is hard to know,
• Identification strategies are stringent
• My sense is we’re over-correcting on this front; let’s figure out what’s
there first.
Selection
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Y
X1 X2
Weak instruments bias us toward null effects
Y
X1 X2
I
Possible solutions:
• Theory: Given what we know about how friendships form, is it reasonable
to assume a bi-directional cause? That is, work through the meeting,
socializing, etc. process and ask whether it makes sense that Y is a cause of
W. This will not convince a skeptical reader, but you should do it anyway.
• Models:
- Time Order. Necessary but not sufficient. We are on somewhat firmer
ground if W precedes Y in time, but the Shalizi & Thomas problem of
an as-yet-earlier joint confounder is still there.
- Simultaneous Models. Model both the friendship pattern and the
outcome of interest simultaneously. Best bet for direct estimation
•Sensitivity Analysis:
I think the most reasonable solution…take error potential seriously,
attempt to evaluate how big a problem it really is.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Table 4. Selected SIENA Parameter Estimates: Parental Knowledge, Parental Discipline, and Drinking
a
Model 2
b SE t SD
Selection parameters
Alter effects: Who is more often named as a friend?
Parental knowledge -0.002 0.004 -0.47 0.003
Parental discipline -0.004 0.002 -1.55 0.001
Drinking 0.083 0.010 8.69 *** 0.007
Ego effects: Who names more friends?
Parental knowledge 0.044 0.007 5.85 *** 0.039
Parental discipline -0.002 0.005 -0.36 0.023
Drinking -0.011 0.021 -0.53 0.089
Similarity effects: Choosing friends similar to oneself
Parental knowledge 0.169 0.025 6.70 *** 0.101
Parental discipline 0.151 0.017 8.86 *** 0.035
Drinking 0.276 0.021 13.45 *** 0.006
Behavioral parameters: Influence on Drinking
Friends' attributes
Mean Parental knowledge -0.230 0.065 -3.56 ** 0.014
Mean Parental discipline -0.051 0.043 -1.18 0.011
Drinking mean similarity 1.162 0.110 10.56 *** 0.023
Control variables (individual level)
Parental knowledge -0.122 0.014 -8.93 *** 0.004
Parental discipline -0.043 0.009 -4.54 *** 0.002
***p < .001. **p < .01. *p < .05. †p < .10.
a
Models also include rate and shape parameters, structural parameters, and the full set of alter,
ego, similarity, and individual-level control parameters
SIENA model of drinking
Daniel T. Ragan, D. Wayne Osgood
Possible solutions:
•Sensitivity Analysis:
I think the most
reasonable solution…take
error potential seriously,
attempt to evaluate how
big a problem it really is.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Possible solutions:
•Sensitivity Analysis:
I think the most reasonable solution…take error potential seriously, attempt to evaluate
how big a problem it really is.
Network Diffusion & Peer Influence
Peer Influence & Health: Current Lit & Controversies
Sociological Methods & Research 2000
04 Diffusion and Peer Influence (2016)

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04 Diffusion and Peer Influence (2016)

  • 1. Diffusion and Peer Influence
  • 2. “you are who you associate with” Time Diffusion & Peer Influence
  • 3. 1. Diffusion A. Compartmental Models B. Network Diffusion i. Topology ii. Timing iii. Structural Transmission a. Complex contagion 2. Peer Influence
  • 4. Coleman, Katz and Menzel, “Diffusion of an innovation among physicians” Sociometry (1957) Our substantive interest in networks is often in how things move through them, from disease to ideas to behavior. Network Diffusion & Peer Influence Basics
  • 5. Figure 2. Binge Drinking Predicted Probabilities by Gender and Friends’ Prior Drinking. Derek A. Kreager, and Dana L. Haynie American Sociological Review 2011;76:737-763 Romantic partnerships in high school lead to adoption of partners’ friends’ behaviors Network Diffusion & Peer Influence Basics
  • 6. Table 4. Proportion of Ties Created or Maintained Over Time by Network Process and Depression Level. David R. Schaefer et al. American Sociological Review 2011;76:764-785 Health effects on ties Depressed students form ties through non-normative network processes
  • 7. Network Diffusion & Peer Influence Basics Classic (disease) diffusion makes use of compartmental models. Large N and homogenous mixing allows one to express spread as generalized probability models. Works very well for highly infectious bits in large populations… SI(S) model – actors are in only two states, susceptible or infectious. See: https://wiki.eclipse.org/Introduction_to_Compartment_Models for general introduction. SIIR(S) model – adds an “exposed” but not infectious state and recovered.
  • 8. Network Diffusion & Peer Influence Basics – might even help understand the zombie apocalypse http://loe.org/images/content/091023/Zombie%20Publication.pdf
  • 9. Network Diffusion & Peer Influence Basics Network Models Same basic SI(R,Z,etc) setup, but connectivity is not assumed random, rather it is structured by the network contact pattern. If pij is small or the network is very clustered, these two can yield very different diffusion patterns.* Real Random *these conditions do matter. Compartmental models work surprisingly well if the network is large, dense or the bit highly infectiousness…because most networks have a bit of randomness in them. We are focusing on the elements that are unique/different for network as opposed to general diffusion.
  • 10. Network Diffusion & Peer Influence Basics If 0 < pij < 1
  • 11. Network Diffusion & Peer Influence Basics If 0 < pij < 1 0.01 0.06 0.11 0.26 0.46
  • 12. In addition to* the dyadic probability that one actor passes something to another (pij), two factors affect flow through a network: Topology - the shape, or form, of the network - Example: one actor cannot pass information to another unless they are either directly or indirectly connected Time - the timing of contact matters - Example: an actor cannot pass information he has not receive yet *This is a big conditional! – lots of work on how the dyadic transmission rate may differ across populations. Key Question: What features of a network contribute most to diffusion potential? Network Diffusion & Peer Influence Network diffusion features Use simulation tools to explore the relative effects of structural connectivity features
  • 13. • A network has to be connected for a bit to pass over it • If transmission is uncertain, the longer the distance the lower the likelihood of spread. 0 0.2 0.4 2 3 4 5 6 Path distance probability Distance and diffusion (p(transfer)=pij dist Here pij of 0.6 Network Diffusion & Peer Influence Network diffusion features We need: (1) reachability (2) distance (3) local clustering (4) multiple routes (5) star spreaders
  • 14. • Local clustering turns flow “in” on a potential transmission tree Arcs: 11 Largest component: 12, Clustering: 0 Arcs: 11 Largest component: 8, Clustering: 0.205 We need: (1) reachability (2) distance (3) local clustering (4) multiple routes (5) star spreaders Network Diffusion & Peer Influence Network diffusion features
  • 15. • The more alternate routes one has for transmission, the more likely flow should be. • Operationalize alternate routes with structural cohesion We need: (1) reachability (2) distance (3) local clustering (4) multiple routes (5) star spreaders Network Diffusion & Peer Influence Network diffusion features
  • 16. Probability of transfer by distance and number of non-overlapping paths, assume a constant pij of 0.6 0 0.2 0.4 0.6 0.8 1 1.2 2 3 4 5 6 Path distance probability 1 path C P X Y 10 paths 5 paths 2 paths Cohesion  Redundancy Diffusion Network Diffusion & Peer Influence Network diffusion features
  • 17. 0 1 2 3 Node Connectivity As number of node-independent paths C P X Y Structural Cohesion: A network’s structural cohesion is equal to the minimum number of actors who, if removed from the network, would disconnect it. Network Diffusion & Peer Influence Network diffusion features
  • 18. STD Transmission danger: sex or drugs? Structural core more realistic than nominal core C P X Y Data from “Project 90,” of a high-risk population in Colorado Springs Network Diffusion & Peer Influence Network diffusion features
  • 19. • Much of the work on “core groups” or “at risk” populations focus on high-degree nodes. The assumption is that high-degree nodes are likely to contact lots of people. We need: (1) reachability (2) distance (3) local clustering (4) multiple routes (5) star spreaders Network Diffusion & Peer Influence Network diffusion features
  • 20. • Much of the work on “core groups” or “at risk” populations focus on high-degree nodes. The assumption is that high-degree nodes are likely to contact lots of people. We need: (1) reachability (2) distance (3) local clustering (4) multiple routes (5) star spreaders Network Diffusion & Peer Influence Network diffusion features
  • 21. Network Diffusion & Peer Influence Network diffusion features Assortative mixing: A more traditional way to think about “star” effects.
  • 22. • Simulation study: How do these different features compare over a collection of observed nets? • For each network trial: • Fix dyadic transmission probability • Randomly select a node as seed • Trace the diffusion path across the network • Measure speed & extent of spread • Model extent of spread by structural characteristics • First run: Add Health: • simple diffusion process • dyadic probability set to 0.08 • 500 trials in each network. • Then expand to : • Different assumptions of dyadic transmission probability • Different data set (Facebook) • Complex diffusion models Network Diffusion & Peer Influence Network diffusion features: simulation test
  • 23. Network Diffusion & Peer Influence Network diffusion features: simulation test
  • 24. Network Diffusion & Peer Influence Network diffusion features: simulation test
  • 25. Define as a general measure of the “diffusion susceptibility” of a graph’s structure as the ratio of the area under the observed curve to the area under the curve for a matching random network. As this gets smaller than 1.0, you get effectively slower median transmission. Network Diffusion & Peer Influence Network diffusion features: simulation test
  • 26. Table 2. OLS Regression of Relative Diffusion Ratio on Network Structure Variable Model 1 Model 2 Model 3 Model 4 Model 5 Intercept 1.62*** 1.90*** 1.02*** 1.81*** 1.71*** Connectivity Distance -0.207*** -0.179*** -0.171*** Independent Paths -0.077*** -0.056*** -0.052*** Distance x Paths 0.023*** 0.015*** 0.016*** Clustering Clustering Coefficient -0.692*** -0.653*** -0.454*** Grade Homophily -0.026** -0.007 -0.009* Peer Group Strength -0.868*** -0.141 -0.146 Degree Distribution Degree Skew -0.023 -0.007 -0.002 Assortative Mixing -0.189* -0.059 -0.071 Control Variables Network Size/100 0.005*** -0.005*** -.005*** 0.004* 0.002** Proportion Isolated -0.007 -1.106*** -.984*** -0.300* 0.058 Non-Complete -0.006 -0.052* -.078** -0.006 0.018 Adj- R2 0.85 0.76 0.60 0.90 0.93 N 124 124 124 124 121 Network Diffusion & Peer Influence Network diffusion features: simulation test
  • 27. Table 2. OLS Regression of Relative Diffusion Ratio on Network Structure Variable Model 1 Model 2 Model 3 Model 4 Model 5 Intercept 1.62*** 1.90*** 1.02*** 1.81*** 1.71*** Connectivity Distance -0.207*** -0.179*** -0.171*** Independent Paths -0.077*** -0.056*** -0.052*** Distance x Paths 0.023*** 0.015*** 0.016*** Clustering Clustering Coefficient -0.692*** -0.653*** -0.454*** Grade Homophily -0.026** -0.007 -0.009* Peer Group Strength -0.868*** -0.141 -0.146 Degree Distribution Degree Skew -0.023 -0.007 -0.002 Assortative Mixing -0.189* -0.059 -0.071 Control Variables Network Size/100 0.005*** -0.005*** -.005*** 0.004* 0.002** Proportion Isolated -0.007 -1.106*** -.984*** -0.300* 0.058 Non-Complete -0.006 -0.052* -.078** -0.006 0.018 Adj- R2 0.85 0.76 0.60 0.90 0.93 N 124 124 124 124 121 Network Diffusion & Peer Influence Network diffusion features: simulation test
  • 28. Table 2. OLS Regression of Relative Diffusion Ratio on Network Structure Variable Model 1 Model 2 Model 3 Model 4 Model 5 Intercept 1.62*** 1.90*** 1.02*** 1.81*** 1.71*** Connectivity Distance -0.207*** -0.179*** -0.171*** Independent Paths -0.077*** -0.056*** -0.052*** Distance x Paths 0.023*** 0.015*** 0.016*** Clustering Clustering Coefficient -0.692*** -0.653*** -0.454*** Grade Homophily -0.026** -0.007 -0.009* Peer Group Strength -0.868*** -0.141 -0.146 Degree Distribution Degree Skew -0.023 -0.007 -0.002 Assortative Mixing -0.189* -0.059 -0.071 Control Variables Network Size/100 0.005*** -0.005*** -.005*** 0.004* 0.002** Proportion Isolated -0.007 -1.106*** -.984*** -0.300* 0.058 Non-Complete -0.006 -0.052* -.078** -0.006 0.018 Adj- R2 0.85 0.76 0.60 0.90 0.93 N 124 124 124 124 121 Network Diffusion & Peer Influence Network diffusion features: simulation test
  • 29. Figure 4. Relative Diffusion Ratio By Distance and Number of Independent Paths 0.4 0.6 0.8 1 1.2 2.3 2.8 3.3 3.8 4.3 4.8 5.3 5.8 6.3 Average Path Length Observed/Random k=2 k=4 k=6 k=8 Figure 4. Relative Diffusion Ratio By Distance and Number of Independent Paths 0.4 0.6 0.8 1 1.2 2.3 2.8 3.3 3.8 4.3 4.8 5.3 5.8 6.3 Average Path Length Observed/Random k=2 k=4 k=6 k=8 Network Diffusion & Peer Influence Network diffusion features: simulation test
  • 30. Traditional “core group” models have a local-vision understanding of risk: those with lots of ties (high degree) are the focus for intervention and actions. •In the short time-windows necessary for STD transfer, low-degree networks are the relevant features for transmission. What sorts of networks emerge when average degree (in the short run) is held to small numbers? •How does the shape of the degree distribution matter? If activity is homogeneous do we get fundamentally different networks than if it is very heterogeneous? Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 31. Partner Distribution Component Size/Shape Emergent Connectivity in low-degree networks Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 32. Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 33. In both distributions, a giant component & reconnected core emerges as density increases, but at very different speeds and ultimate extent. Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 34. What distinguishes these two distributions? Shape Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 35. What distinguishes these two distributions? Shape: The scale-free network’s signature is the long-tail So what effect does changes in the shape have on connectiv Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 36.  Volume  DispersionxSkewness Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 37. Search Procedure: 1) Identify all valid degree distributions with the given mean degree and a maximum of 6 w. brute force search. 2) Map them to this space 3) Simulate networks each degree distribution 4) Measure size of components & Bicomponents Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 38. Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 39. Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 40. C:45%, B: 8.5% Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 41. C:45%, B: 8.5% Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 42. C:83%, B: 36% Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 43. Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), C:99%, B: 86% Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 44. Largest Component (at least 1 path) Largest Bicomponent (at least 2 paths) Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), Network Diffusion & Peer Influence A closer look at emerging connectivity
  • 45. In addition to* the dyadic probability that one actor passes something to another (pij), two factors affect flow through a network: Topology - the shape, or form, of the network - Example: one actor cannot pass information to another unless they are either directly or indirectly connected Time - the timing of contact matters - Example: an actor cannot pass information he has not receive yet *This is a big conditional! – lots of work on how the dyadic transmission rate may differ across populations. Key Question: What features of a network contribute most to diffusion potential? Network Diffusion & Peer Influence Relational Dynamics Use simulation tools to explore the relative effects of structural connectivity features
  • 46. Three relevant networks Discussions of network effects on STD spread often speak loosely of “the network.” There are three relevant networks that are often conflated: 1) The contact network. The set of pairs of people connected by sexual contact. G(V,E). 2) The exposure network. A subset of the edges in the contact network where timing makes it possible for one person to pass infection to another. 3) The transmission network. The subset of the exposure network where disease is actually passed. In most cases this is a tree layered on (2) and rooted on a source/seed node. Network Diffusion & Peer Influence Relational Dynamics
  • 47. Contact network: Everyone, it is a connected component Who can “A” reach? Network Diffusion & Peer Influence Relational Dynamics Discussions of network effects on STD spread often speak loosely of “the network.” There are three relevant networks that are often conflated: Three relevant networks
  • 48. Exposure network: here, node “A” could reach up to 8 others Who can “A” reach? Network Diffusion & Peer Influence Relational Dynamics Discussions of network effects on STD spread often speak loosely of “the network.” There are three relevant networks that are often conflated: Three relevant networks
  • 49. Transmission network: upper limit is 8 through the exposure links (dark blue). Transmission is path dependent: if no transmission to B, then also none to {K,L,O,J,M} Who can “A” reach? Exposable Link (from A’s p.o.v.) Contact Network Diffusion & Peer Influence Relational Dynamics Discussions of network effects on STD spread often speak loosely of “the network.” There are three relevant networks that are often conflated: Three relevant networks
  • 50. The mapping between the contact network and the exposure network is based on relational timing. In a dynamic network, edge timing determines if something can flow down a path because things can only be passed forward in time. Definitions: Two edges are adjacent if they share a node. A path is a sequence of adjacent edges (E1, E2, …Ed). A time-ordered path is a sequence of adjacent edges where, for each pair of edges in the sequence, the start time Si is less than or equal to Ej S(E1) < E(E2) Adjacent edges are concurrent if they share a node and have start and end dates that overlap. This occurs if: S(E2) < E(E1) Concurrency Network Diffusion & Peer Influence Relational Dynamics
  • 51. A B C D time 1 2 3 4 5 6 7 8 9 10 AB BC CE E CD 2 - 71 - 3 S(ab) E(ab) S(bc) E(bc) S(ce) E(ce) The mapping between the contact network and the exposure network is based on relational timing. In a dynamic network, edge timing determines if something can flow down a path because things can only be passed forward in time. Concurrency Network Diffusion & Peer Influence Relational Dynamics
  • 52. The constraints of time-ordered paths change our understanding of the system structure of the network. Paths make a network a system: linking actors together through indirect connections. Relational timing changes how paths cumulate in networks. Indirect connectivity is no longer transitive: A B C D1 - 2 3 - 4 1 - 2 Here A can reach C, and C and reach D. But A cannot reach D (nor D A). Why? Because any infection A passes to C would have happened after the relation between C and D ended. A B C D1 - 2 3 - 4 1 - 2 Network Diffusion & Peer Influence Relational Dynamics
  • 53. Edge time structures are characterized by sequence, duration and overlap. Paths between i and j, have length and duration, but these need not be symmetric even if the constituent edges are symmetric. Network Diffusion & Peer Influence Relational Dynamics
  • 54. 1 2 2 2 2 2 1 1 1 2 2 2 2 2 2 1 1 2 2 2 2 2 2 1 1 2 2 2 2 2 2 1 1 2 2 2 2 2 2 1 1 2 2 2 2 2 2 1 1 1 2 2 2 2 2 1 Implied Contact Network of 8 people in a ring All relations Concurrent Reachability = 1.0 Network Diffusion & Peer Influence Relational Dynamics
  • 55. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Implied Contact Network of 8 people in a ring Serial Monogamy (1) 1 2 3 7 6 5 8 4 Reachability = 0.71 Network Diffusion & Peer Influence Relational Dynamics
  • 56. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Implied Contact Network of 8 people in a ring Mixed Concurrent 2 2 1 1 2 2 3 3 Reachability = 0.57 Network Diffusion & Peer Influence Relational Dynamics
  • 57. Implied Contact Network of 8 people in a ring Serial Monogamy (3) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 2 2 Reachability = 0.43 Network Diffusion & Peer Influence Relational Dynamics
  • 58. 1 2 1 1 2 1 2 2 Timing alone can change mean reachability from 1.0 when all ties are concurrent to 0.42. In general, ignoring time order is equivalent to assuming all relations occur simultaneously – assumes perfect concurrency across all relations. Network Diffusion & Peer Influence Relational Dynamics
  • 59. A B C D E F A 0 1 2 2 4 1 B 1 0 1 2 3 2 C 0 1 0 1 2 2 D 0 0 1 0 1 1 E 0 0 0 1 0 2 F 1 0 0 1 0 0 While a is 2 steps from d, and d is 1 step from e, a and e are 4 steps apart. This is because the shorter path from a to e emerges after the path from d to e ended. 4 2 1 Path distances no longer simply add Network Diffusion & Peer Influence Relational Dynamics
  • 60. A B C D 3 - 4 E The geodesic from A to D is AE, ED and is two steps long. But the fastest path would be AB, BC, CD, which while 3 steps long could get there by time 5 compared to time 7. I ignore this feature for the remainder… Aside: “shortest” can now be replaced with “fastest” Network Diffusion & Peer Influence Relational Dynamics
  • 61. Concurrency affects exposure by making paths symmetric, which increases exposure down multiple “branches” of a contact sequence. Consider a simplified example: All edges concurrent a b c d e f g h i j k l m a . 1 1 1 1 1 1 1 1 1 1 1 1 b 1 . 1 1 1 1 1 1 1 1 1 1 1 c 1 1 . 1 1 1 1 1 1 1 1 1 1 d 1 1 1 . 1 1 1 1 1 1 1 1 1 e 1 1 1 1 . 1 1 1 1 1 1 1 1 f 1 1 1 1 1 . 1 1 1 1 1 1 1 g 1 1 1 1 1 1 . 1 1 1 1 1 1 h 1 1 1 1 1 1 1 . 1 1 1 1 1 i 1 1 1 1 1 1 1 1 . 1 1 1 1 j 1 1 1 1 1 1 1 1 1 . 1 1 1 k 1 1 1 1 1 1 1 1 1 1 . 1 1 l 1 1 1 1 1 1 1 1 1 1 1 . 1 m 1 1 1 1 1 1 1 1 1 1 1 1 . Network Diffusion & Peer Influence Relational Dynamics
  • 62. All edges except gd concurrent a b c d e f g h i j k l m a . 1 1 1 1 1 1 1 1 1 1 1 1 b 1 . 1 1 1 1 1 1 1 1 1 1 1 c 1 1 . 1 1 1 1 1 1 1 1 1 1 d 1 1 1 . 1 1 1 1 1 1 1 1 1 e 1 1 1 1 . 1 1 1 1 1 1 1 1 f 1 1 1 1 1 . 1 1 1 1 1 1 1 g 0 0 0 1 0 0 . 1 1 1 1 1 1 h 0 0 0 1 0 0 1 . 1 1 1 1 1 i 0 0 0 1 0 0 1 1 . 1 1 1 1 j 0 0 0 1 0 0 1 1 1 . 1 1 1 k 0 0 0 1 0 0 1 1 1 1 . 1 1 l 0 0 0 1 0 0 1 1 1 1 1 . 1 m 0 0 0 1 0 0 1 1 1 1 1 1 . Concurrency affects exposure by making paths symmetric, which increases exposure down multiple “branches” of a contact sequence. Consider a simplified example: Network Diffusion & Peer Influence Relational Dynamics
  • 63. Concurrency affects exposure by making paths symmetric, which increases exposure down multiple “branches” of a contact sequence. Consider a simplified example: All edges gd after all others: a b c d e f g h i j k l m a . 1 1 1 1 1 1 0 0 0 0 0 0 b 1 . 1 1 1 1 1 0 0 0 0 0 0 c 1 1 . 1 1 1 1 0 0 0 0 0 0 d 1 1 1 . 1 1 1 0 0 0 0 0 0 e 1 1 1 1 . 1 1 0 0 0 0 0 0 f 1 1 1 1 1 . 1 0 0 0 0 0 0 g 0 0 0 1 0 0 . 1 1 1 1 1 1 h 0 0 0 0 0 0 1 . 1 1 1 1 1 i 0 0 0 0 0 0 1 1 . 1 1 1 1 j 0 0 0 0 0 0 1 1 1 . 1 1 1 k 0 0 0 0 0 0 1 1 1 1 . 1 1 l 0 0 0 0 0 0 1 1 1 1 1 . 1 m 0 0 0 0 0 0 1 1 1 1 1 1 . The concurrency status of {dg} determines which “side” of the graph is exposed. Note this effect happens at the system level – the correlation between exposure and node-level timing is essentially zero (d/g excepted) Network Diffusion & Peer Influence Relational Dynamics
  • 64. Resulting infection trace from a simulation (Morris et al, AJPH 2010). Observed infection paths from 10 seeds in an STD simulation, edges coded for concurrency status. Network Diffusion & Peer Influence Relational Dynamics
  • 65. Resulting infection trace from a simulation (Morris et al, AJPH 2010). Network Diffusion & Peer Influence Relational Dynamics Observed infection paths from 10 seeds in an STD simulation, edges coded for concurrency status.
  • 66. Timing constrains potential diffusion paths in networks, since bits can flow through edges that have ended. This means that: • Structural paths are not equivalent to the diffusion-relevant path set. • Network distances don’t build on each other. • Weakly connected components overlap without diffusion reaching across sets. • Small changes in edge timing can have dramatic effects on overall diffusion • Diffusion potential is maximized when edges are concurrent and minimized when they are “inter-woven” to limit reachability. Combined, this means that many of our standard path-based network measures will be incorrect on dynamic graphs. Network Diffusion & Peer Influence Relational Dynamics
  • 67. The distribution of paths is important for many of the measures we typically construct on networks, and these will be change if timing is taken into consideration: Centrality: Closeness centrality Path Centrality Information Centrality Betweenness centrality Network Topography Clustering Path Distance Groups & Roles: Correspondence between degree-based position and reach-based position Structural Cohesion & Embeddedness Opportunities for Time-based block-models (similar reachability profiles) In general, any measures that take the systems nature of the graph into account will differ. Structural measurement implications Network Diffusion & Peer Influence Relational Dynamics – other implication of dynamic nets (briefly)
  • 68. New versions of classic reachability measures: 1) Temporal reach: The ij cell = 1 if i can reach j through time. 2) Temporal geodesic: The ij cell equals the number of steps in the shortest path linking i to j over time. 3) Temporal paths: The ij cell equals the number of time-ordered paths linking i to j. These will only equal the standard versions when all ties are concurrent. Duration explicit measures 4) Quickest path: The ij cell equals the shortest time within which i could reach j. 5) Earliest path: The ij cell equals the real-clock time when i could first reach j. 6) Latest path: The ij cell equals the real-clock time when i could last reach j. 7) Exposure duration: The ij cell equals the longest (shortest) interval of time over which i could transfer a good to j. Each of these also imply different types of “betweenness” roles for nodes or edges, such as a “limiting time” edge, which would be the edge whose comparatively short duration places the greatest limits on other paths. Structural measurement implications Network Diffusion & Peer Influence Relational Dynamics – other implication of dynamic nets (briefly)
  • 69. Topology & Time interact: How relational sequencing affects diffusion is conditioned by the structural patterns of relations. Examples: - Time limitations mean star nodes can’t interact with everyone at each time tick; effects of high degree are thus limited by schedule/availabiltiy - If within-cluster ties are also more frequent than between cluster ties, then the effects of communities will be magnified. -Multiple connectivity should provide routes around breaks built by temporal sequence. Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  • 70. Measures Dependent variable: Reachability in the exposure graph. This is the proportion of pairs in the network that are reachable in time. Exposure Graph Density Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  • 71. Measures: Independent variables Features of the topology. Of key interest is the level of structural cohesion. 1 2 3 4 5 6 7 8 9 1 -- 1 1 1 1 1 1 1 1 2 1 -- 2 2 2 1 1 1 1 3 1 2 -- 2 2 1 1 1 1 4 1 2 2 -- 2 1 1 1 1 5 1 2 2 2 -- 1 1 1 1 6 1 1 1 1 1 -- 2 2 1 7 1 1 1 1 1 2 -- 2 1 8 1 1 1 1 1 2 2 -- 1 9 1 1 1 1 1 1 1 1 -- Cell Value = highest k-connected component pair belongs to. Average = 1.25 Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  • 72. 1 2 3 4 5 6 7 8 9 1 -- 1 1 1 1 1 1 1 1 2 1 -- 3 3 3 2 2 2 1 3 1 3 -- 3 3 2 2 2 1 4 1 3 3 -- 3 2 2 2 1 5 1 3 3 3 -- 2 2 2 1 6 1 2 2 2 2 -- 2 2 1 7 1 2 2 2 2 2 -- 2 1 8 1 2 2 2 2 2 2 -- 1 9 1 1 1 1 1 1 1 1 -- Cell Value = highest k-connected component pair belongs to. Average = 1.6 Measures: Independent variables Features of the topology. Of key interest is the level of structural cohesion. Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  • 73. Measures: Independent variables Features of the topology. Of key interest is the level of structural cohesion. Average connectivity Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  • 74. Volume Distance Connectivity Nodes: 148 Mean Deg: 6.16 Density: 0.042 Centralization: 0.187 Nodes: 80 Mean Deg: 5.27 Density: 0.067 Centralization: 0.373 Nodes: 154 Mean Deg: 3.71 Density: 0.025 Centralization: 0.147 Nodes: 128 Mean Deg: 3.39 Density: 0.027 Centralization: 0.205 Mean: 3.59 Diameter: 5 Centralization: 0.312 Mean: 3.02 Diameter: 5 Centralization: 0.413 Mean: 4.99 Diameter: 8 Centralization: 0.259 Mean: 4.55 Diameter: 6 Centralization: 0.301 Largest BC: 0.51 Pairwise K: 1.57 Largest BC: 0.33 Pairwise K: 1.34 Largest BC: 0.08 Pairwise K: 1.07 Largest BC: Pairwise K: 1.06 Exemplar independent variables “HighCohesive”“LowCohesive”Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  • 75. Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  • 76. “Low Cohesive” Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  • 77. Proportion of relations concurrent DensityoftheExposureNetwork Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  • 78. Colors=different nets Panels=cohesion level Proportion of relations concurrent DensityoftheExposureNetwork Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  • 79. Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  • 80. 1. Concurrency has a necessarily positive effect on potential diffusion exposure 1. This implies that we should see greater transmission given greater concurrency 2. This works by creating “multiple routes” in the exposure path structure 2. Structural cohesion captures multiple routes in the contact graph 1. Higher levels of cohesion increase exposure by directly increasing the underlying transmission substrate 3. There is a negative interaction between cohesion and concurrency: as cohesion increases, the relative returns to concurrency decrease. 1. But this comes at the cost of a higher base-level of exposure. Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  • 81. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Thus far we have focused on a “simple” dyadic diffusion parameter, pij, where the probability of passing/receiving the bit is purely dependent on discordant status of the dyad, sometimes called the “independent cascade model” (), which suggests a monotonic relation between the number of times you are exposed through peers. High exposure could be due to repeated interaction with one person or weak interaction with many, effectively equating: Alternative models exist. Under “complex contagion” for example, the likelihood that I accept the bit that flows through the network depends on the proportion of my peers that have the bit.
  • 82. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion 1 1 2 3 Complex Contagion Assume adoption requires k neighbors having adopted, then transmission can only occur within dense clusters:
  • 83. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Assume adoption requires k neighbors having adopted, then transmission can only occur within dense clusters: Assume pij=1, k=2, starting nodes in yellow
  • 84. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Assume adoption requires k neighbors having adopted, then transmission can only occur within dense clusters: For this network under weak complex diffusion (k=2), the maximum risk size is 8.
  • 85. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Assume adoption requires k neighbors having adopted, then transmission can only occur within dense clusters: For this network under weak complex diffusion (k=2), the maximum risk size is reaches 98%. One of the Prosper schools: Start
  • 86. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Can lead to widely varying sizes of potential diffusion cascades. Here’s the distribution across all PROPSPER schools: Distribution is largely bimodal (even with a connected pair start)
  • 87. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Can lead to widely varying sizes of potential diffusion cascades. Here’s the distribution across all PROPSPER schools: The governing factors are (a) curved effect of local redundancy and (b) structural cohesion Network Average Proportion Reached k=2 complex contagion MeanCascadeSize Coh=0.3 Coh=1.2 Coh=2.2 Coh=3.2 Coh=4.1
  • 88. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Does get used for real health work: Here , authors assume a CC process, seeded with observed depressive cases, turn that into a Markov model and ask what parameters would maximize fit from simulated to observed.
  • 89. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex diffusion is just the most well studied of the options that combine transmission with some pairwise positional feature. This is a wide-open area for future research. The basic idea is that transmission is increased/decreased if there is some third structural property that the susceptible & infected pair share. This leads us into the general problem of peer influence models…when do peers change each other’s behaviors?
  • 90. Background: • Long standing research interest in how our relations shape our attitudes and behaviors. • Most often assumed mechanism is that people (through conversation or similar) change each others beliefs/opinions, which changes behavior. This implies that position in a communication network should be related to attitudes. • Alternatives: • Modeling behavior: ego copies behavior of alter to gain respect, esteem, etc. • Distinction: Ego tries to be different from (some) alter to gain respect, esteem, etc. • Access: Ego wants to do Y, but can only do so because alter provides access (say, being old enough to buy cigarettes). Network Diffusion & Peer Influence Peer Influence Dynamics
  • 91. Background: • Early work was ego-centric – people informed on their peers •Seems to have inflated PI effects by ~50% or so…either through projection of ego behavior onto peers or selective interaction (what alters do with ego may be different than what alter does all the time). •Then to cross sectional associations based on alter self-reports •Better, but still likely conflates selection with influence •Next to dynamic models: •Ego Behavior(t) ~ f(ego behavior(t-1) + alter behavior (t-1) + controls •Much better; still debate on (a) correct estimation functions, (b) unobserved selection features that confound causal inference. •Development of Actor-oriented models (SIENA) Network Diffusion & Peer Influence Peer Influence Dynamics
  • 92. Background: •Finally: Experimental manipulation of peer exposure •“Gold standard” for isolation of peer effects •Likely strongly underestimates effects (as measure intent to treat, not take- up of treatment, since people may not care about relations that can be manipulated). b(Peer(y)): Ego Inform < Alter Inform < Cross Sectional < Dynamic < Experimental. Still often find peer effects, but my sense is that we’ve (strongly) over-corrected at this point. Network Diffusion & Peer Influence Peer Influence Dynamics
  • 93. Freidkin’s Structural Theory of Social Influence : Two-part model: Beliefs are a function of two sources: a) Individual characteristics •Gender, Age, Race, Education, Etc. Standard sociology b) Interpersonal influences •Actors negotiate with others Network Diffusion & Peer Influence Peer Influence Dynamics
  • 94. XBY )1( (1) )1()1()( )1( YWYY αα Tt   (2) Y(1) = an N x M matrix of initial opinions on M issues for N actors X = an N x K matrix of K exogenous variable that affect Y B = a K x M matrix of coefficients relating X to Y a = a weight of the strength of endogenous interpersonal influences W = an N x N matrix of interpersonal influences Network Diffusion & Peer Influence Peer Influence Dynamics
  • 95. XBY )1( (1) This is the standard sociology model for explaining anything: the General Linear Model. It says that a dependent variable (Y) is some function (B) of a set of independent variables (X). At the individual level, the model says that:  k kiki BXY Usually, one of the X variables is e, the model error term. Network Diffusion & Peer Influence Peer Influence Dynamics
  • 96. )1()1()( )1( YWYY αα Tt   (2) This part of the model taps social influence. It says that each person’s final opinion is a weighted average of their own initial opinions )1( )1( Yα And the opinions of those they communicate with (which can include their own current opinions) )1( T αWY Network Diffusion & Peer Influence Peer Influence Dynamics
  • 97. The key to the peer influence part of the model is W, a matrix of interpersonal weights. W is a function of the communication structure of the network, and is usually a transformation of the adjacency matrix. In general:    j ij ij w w 1 10 Various specifications of the model change the value of wii, the extent to which one weighs their own current opinion and the relative weight of alters. Network Diffusion & Peer Influence Peer Influence Dynamics
  • 98. 1 2 3 4 1 2 3 4 1 1 1 1 0 2 1 1 1 0 3 1 1 1 1 4 0 0 1 1 1 2 3 4 1 .33 .33 .33 0 2 .33 .33 .33 0 3 .25 .25 .25 .25 4 0 0 .50 .50 1 2 3 4 1 .50 .25 .25 0 2 .25 .50 .25 0 3 .20 .20 .40 .20 4 0 0 .33 .67 Even 2*self 1 2 3 4 1 .50 .25 .25 0 2 .25 .50 .25 0 3 .17 .17 .50 .17 4 0 0 .50 .50 degree Self weight: 1 2 3 4 1 2 1 1 0 2 1 2 1 0 3 1 1 2 1 4 0 0 1 2 1 2 3 4 1 2 1 1 0 2 1 2 1 0 3 1 1 3 1 4 0 0 1 1 Network Diffusion & Peer Influence Peer Influence Dynamics
  • 99. )1()1()( )1( YWYY αα Tt   Formal Properties of the model When interpersonal influence is complete, model reduces to: )1( )1()1()( 01     T Tt WY YWYY When interpersonal influence is absent, model reduces to: )1( )1()1()( 0 Y YWYY   Tt (2) Network Diffusion & Peer Influence Peer Influence Dynamics
  • 100. Formal Properties of the model The model is directly related to spatial econometric models: If we allow the model to run over t and W remains constant: XBWYY )1()()( αα   eb   XWYY ~)()( α Where the two coefficients (a and b) are estimated directly (See Doreian, 1982, SMR). This is the linear network auto correlation model, best bet with cross-sectional data (and randomization trick to estimate se) Network Diffusion & Peer Influence Peer Influence Dynamics
  • 101. Simple example 1 2 3 4 1 2 3 4 1 .33 .33 .33 0 2 .33 .33 .33 0 3 .25 .25 .25 .25 4 0 0 .50 .50 Y 1 3 5 7 a = .8 T: 0 1 2 3 4 5 6 7 1.00 2.60 2.81 2.93 2.98 3.00 3.01 3.01 3.00 3.00 3.21 3.33 3.38 3.40 3.41 3.41 5.00 4.20 4.20 4.16 4.14 4.14 4.13 4.13 7.00 6.20 5.56 5.30 5.18 5.13 5.11 5.10 By t=7, still variability in Y Network Diffusion & Peer Influence Peer Influence Dynamics
  • 102. 1 2 3 4 1 2 3 4 1 .33 .33 .33 0 2 .33 .33 .33 0 3 .25 .25 .25 .25 4 0 0 .50 .50 Y 1 3 5 7 a = 1.0 1.00 3.00 3.33 3.56 3.68 3.74 3.78 3.81 3.00 3.00 3.33 3.56 3.68 3.74 3.78 3.81 5.00 4.00 4.00 3.92 3.88 3.86 3.85 3.84 7.00 6.00 5.00 4.50 4.21 4.05 3.95 3.90 By t=7, almost no variability in Y T: 0 1 2 3 4 5 6 7 Simple example Network Diffusion & Peer Influence Peer Influence Dynamics
  • 103. Extended example: building intuition Consider a network with three cohesive groups, and an initially random distribution of opinions: Network Diffusion & Peer Influence Peer Influence Dynamics
  • 104. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  • 105. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  • 106. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  • 107. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  • 108. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  • 109. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  • 110. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  • 111. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  • 112. Extended example: building intuition Consider a network with three cohesive groups, and an initially random distribution of opinions: Now weight in-group ties higher than between group ties Network Diffusion & Peer Influence Peer Influence Dynamics
  • 113. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations, in-group tie: 2
  • 114.
  • 115.
  • 116.
  • 117.
  • 118.
  • 119. Consider the implications for populations of different structures. For example, we might have two groups, a large orthodox population and a small heterodox population. We can imagine the groups mixing in various levels: Little Mixing Moderate Mixing Heavy Mixing .95 .05 .05 .02 .95 .008 .008 .02 .95 .001 .001 .02 Heterodox: 10 people Orthodox: 100 People Network Diffusion & Peer Influence Peer Influence Dynamics
  • 139. In an unbalanced situation (small group vs large group) the extent of contact can easily overwhelm the small group. Applications of this idea are evident in: •Missionary work (Must be certain to send missionaries out into the world with strong in-group contacts) •Overcoming deviant culture (I.e. youth gangs vs. adults) •This is also the mechanism behind why most youth peer influence is a *good* thing – most youth are well behavior and civic minded…so are exerting positive influences on their peers. Network Diffusion & Peer Influence Peer Influence Dynamics
  • 140. Friedkin (1998) generalizes the model so that alpha varies across people. (1) simply changing a to a vector (A), which then changes each person’s opinion directly (2) by linking the self weight (wii) to alpha. )1()1()( )( YAIAWYY  Tt Were A is a diagonal matrix of endogenous weights, with 0 < aii < 1. A further restriction on the model sets wii = 1-aii This leads to a great deal more flexibility in the theory, and some interesting insights. Consider the case of group opinion leaders with unchanging opinions (I.e. many people have high aii, while a few have low): Network Diffusion & Peer Influence Peer Influence Dynamics
  • 141. Group 1 Leaders Group 2 Leaders Group 3 Leaders Peer Opinion Leaders
  • 147. Further extensions of the model might: • Time dependent a: people likely value other’s opinions more early than later in a decision context • Interact a with XB: people’s self weights are a function of their behaviors & attributes • Make W dependent on structure of the network (weight transitive ties greater than intransitive ties, for example) • Time dependent W: The network of contacts does not remain constant, but is dynamic, meaning that influence likely moves unevenly through the network • And others likely abound…. Network Diffusion & Peer Influence Peer Influence Dynamics
  • 148. There are two common ways to test for peer associations through networks. The first estimates the parameters (a and b) of the network autocorrelation model directly, the second transforms the network into a dyadic model, predicting similarity among actors. eb   XWYY ~)()( α Peer influence model: Network Diffusion & Peer Influence Peer Influence Dynamics This is the linear network autocorrelation model, and as specified, the model makes strong assumptions about equilibrium opinion and static relations.  Some variants on this also expand e to include alternative autocorrelation in the error structure.
  • 149. There are two common ways to test for peer associations through networks. The first estimates the parameters (a and b) of the network autocorrelation model directly, the second transforms the network into a dyadic model, predicting similarity among actors. eb   XWYY ~)()( α Peer influence model: Network Diffusion & Peer Influence Peer Influence Dynamics Note that since WY is a a simple vector -- weighted mean of friends Y -- which can be constructed and added to your GLM model. That is, multiple Y by a W matrix, and run the regression with WY as a new variable, and the regression coefficient is an estimate of a. This is what Doriean calls the QAD estimate of peer influence. It’s wrong, a will be biased, but it’s often not terribly wrong if most obvious selection factors are built int0 X
  • 150. An obvious problem with this specification is that cases are, by definition, not independent, hence “network autocorrelation” terminology. In practice, the QAD approach (perhaps combined with a GLS estimator) results in empirical estimates that are “virtually indistinguishable” from MLE (Doreian et al, 1984) The proper way to estimate the peer equation is to use maximum likelihood estimates, and Doreian gives the formulas for this in his paper, and Carter Butts has implemented in in R with the LNAM procedure. An alternative is to use non-parametric approaches, such as the Quadratic Assignment Procedure, to estimate the effects. Network Diffusion & Peer Influence Peer Influence Dynamics
  • 151. Peer influence through Dyad Models Another way to get at peer influence is not through the level of Y, but by assessing the similarity of connected peers. Recall the simulated example: peer influence is reflected in how close points are to each other. Network Diffusion & Peer Influence Peer Influence Dynamics
  • 152. Peer influence through Dyad Models The model is now expressed at the dyad level as: ij k kkijij eXbAbbY  10 Where Y is a matrix of similarities, A is an adjacency matrix, and Xk is a matrix of similarities on attributes Advantages include ease of specifying relation-specific similarity functions. You can add different features of a relation by adjusting/adding “Aij” variables. Disadvantage is that now in addition to network autocorrelation, you have repeated cases (on both sides). But these can be dealt with using non-parametric modeling & testing techniques (QAP, for example). (which we will go over this afternoon) Network Diffusion & Peer Influence Peer Influence Dynamics
  • 153. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 154. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 155. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 156. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 157. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 158. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 159. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 160. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 161. The network shows significant evidence of weight-homophily Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 162. Effects of peer obesity on ego, by peer type Edge-wise regressions of the form: ControlsEgoAltAltEgo previouspreviousCurrentCurrent  )()()( 321 bbb Ego is repeated for all alters; models include random effects on ego id Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 163. This modeling strategy pools observations on edges and estimates a global effect net of change in ego/alter as a control. Here color is a single ego, number is wave (only 2 egos and 3 waves represented). Effects of peer obesity on ego, by peer type ControlsEgoAltAltEgo previouspreviousCurrentCurrent  )()()( 321 bbb 1 Ego-Current Alter Current 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 Peer Effect Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 164. Alterative specifications include using change- change models and allowing for a random effect of peers. This allows for greater variability in peer effects, and the potential to model differences. ee currentpreviouseCurrentprevious ControlstAltAlEgoEgo bb b   )()( 1 Ego-Current Alter Current 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 b be1 be1 Effects of peer obesity on ego, by peer type Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies ee previouspreviousCurrenteCurrent ControlsEgoAltAltEgo bb bbb   )()()( 32 Or difference models:
  • 165. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Critiques of C&F The C&F studies – of obesity, but also other work on the FHS data – turn on the validity of the causal association. All turn on some issue of model miss-specification, typically: • Can’t truly distinguish a network effect from other sources of common influence • “Selection” (“homophily”) or “Common influence” (“Shared environment”) • The most strident work in this area (Salizi • Statistical errors • Misinterpretation of confidence intervals • Poorly specified/estimated models C&H do a nice job of laying out their responses here: http://jhfowler.ucsd.edu/examining_dynamic_social_networks.pdf and here: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2597062/
  • 166. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Critiques of C&F Cohen-Cole, E. and Fletcher, J. M. (2008). Detecting implausible social network effects in acne, height, and headaches: longitudinal analysis. British Medical Journal 337 a2533. Use the same models as C&F on Add Health to show that things which are theoretically unlikely to be contagious appear to be in this form of model. Note these coefficients are substantially smaller than C&F and only significant at the 0.1 level; and not robust to any sensitivity analysis.
  • 167. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Critiques of C&F Lyons, 2011. 1) C&F claim that differences in directional effects support a PI story: • C& F: While mutual friends and egoalter friends are > 0, alterego is not, means ego is emulating alter. • Lyons notes these CIs overlap too much to make any claim about distinguishing them from each other.
  • 168. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Critiques of C&F Lyons, 2011. 2) Insufficient controls for Homophily • C& F: Use of alter’s lagged Y to control for homophily. Logic is that any feature that selected us to be friends at t-1 would have had it’s effect then. • Lyons notes that current and lagged have opposite signs, which seems suspect, and anyway is an insufficient control. He’s likely right here… 3) Directionality cannot distinguish the source of association • C& F: the ordering: mutual, egoalter, alterego suggests an “esteem” model, where ego copies the behavior of alter. • Lyons argues that we would expect the same logic from a simple “foci” of similarity. I don’t find this argument convincing. 4) Random permutation tests cannot establish 3-degree rule • C& F: Association between alters at 1, 2, 3 degrees of separation are higher than we’d expect by chance, based on a permutation test. • Lyons invalid if the data are incomplete, which they certainly are. I don’t find this argument convincing…data are always incomplete…
  • 169. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Critiques of C&F Lyons, 2011. 5) The models are statistically inconsistent (if not incoherent) • C& F: Use separate models for each type of tie, with random effects on ego. • Lyons notes that these really should be treated as simultaneous equations, with shared error structures and so forth. Doing so (a) leads to unidentified models that must force the estimation of the peer effect to 0. That observed ^0 indicates something amiss. • Strikes me as a bit down in the weeds and I’m not convinced here that he’s critiquing them for what they are really doing (argues there are more equations than data, which is patently not true).
  • 170. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Critiques of C&F Lyons, 2011. My sense is that the strategy C&F took was not fundamentally misguided, but the model specification is probably thin; certainly in the obesity paper – less so in some of the later papers appearing after these debates. Ideally you’d have a much better direct model for selection – perhaps even a separate two-stage model (see the Siena module), but here there are very limited observational controls, which would have been easy to add. In later specifications, they do add fixed effects for ego and still find similar results. Commenting on the debate on SocNET – and a related conclusion that only experiments could provide valid inference – Tom Snijders says: “The logical consequence of this is that we are stuck with imperfect methods. Lyons argues as though only perfect methods are acceptable, and while applauding such lofty ideals I still believe that we should accept imperfection, in life as in science. Progress is made by discussion and improvement of imperfections, not by their eradication.” For a full general discussion, see : https://www.lists.ufl.edu/cgi-bin/wa?A2=ind1106&L=SOCNET&P=R11428
  • 171. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Shalizi & Thomas: PI is *generally* confounded So long as there is an unobserved X that causes both ties and behavior, the effect of peers is unidentified.
  • 172. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Shalizi & Thomas: PI is *generally* confounded Only route out is to make X fully informed (or informing) by an observable Z….but realistically there are few things that (a) cause behavior exclusively without any selection pressure (a) or cause ties exclusively without any influence pressure (b) (though note b is what experimental assignments do) (X causes Z, not Y directly) (X causes A, not Y directly)
  • 173. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Shalizi & Thomas: PI is *generally* confounded Should be noted that this is true for *any* effect – there’s always the potential that an unobserved latent variable is creating a spurious effect; This sort of work argues that the only solution is to use experimental (or, sometimes, propensity score style models)…but that’s simply not always feasible practically. We need to beware of making the best the enemy of the good enough…lest we make no progress at all…
  • 174. Willard Van Quine, professor of philosophy and mathematics emeritus from Harvard University who is regarded as one of the most famous philosophers in the world, wrote his doctoral thesis on a 1927 Remington typewriter, which he still uses. However, he "had an operation on it" to change a few keys to accommodate special symbols. "I found I could do without the second period, the second comma -- and the question mark.” "You don't miss the question mark?” "Well, you see, I deal in certainties." Selection or Influence? Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 175. Is it all selection •What do we know about how friendships form? •Opportunity / focal factors - Being members of the same group - In the same class - On the same team - Members of the same church •Structural Relationship factors - Reciprocity - Social Balance •Behavior Homophily - Smoking - Drinking Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 176. -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Network Model Coefficients, In school Networks Network Diffusion & Peer Influence
  • 177. How to correct this problem? •Essentially, this is an omitted variable problem, and my “solution” has been to identify as many potentially relevant alternative variables as I can find. •The strongest possible correction is to use fixed-effects* models that control for all non-varying individual covariates. These have their own problems… •Dual model for influence & selection. •Two-stage model “Heckman” sorts of models •Dynamic SAOM models Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies *“Adding fixed effects to dynamic panel models with many subjects and few repeat observations creates severe bias towards zero coefficients. This has been demonstrated both analytically (Nickell 1981) and through simulations (Nerlove 1971) for OLS and other regression models and has been well-known by social scientists, including economists, for a very long time. In fact, CCF even note that they do not add fixed effects to their logit regression model for this reason, but they strangely assert that fixed effects are necessary in the OLS model.” Estimating Peer Effects on Health in Social Networks : A Response to Cohen-Cole and Fletcher; Trogdon, Nonnemaker, Pais J.H. Fowler, PhD and N.A. Christakis, MD, PhD
  • 178. • Causal status of such similarity is hard to know, • Identification strategies are stringent • My sense is we’re over-correcting on this front; let’s figure out what’s there first. Selection Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Y X1 X2 Weak instruments bias us toward null effects Y X1 X2 I
  • 179. Possible solutions: • Theory: Given what we know about how friendships form, is it reasonable to assume a bi-directional cause? That is, work through the meeting, socializing, etc. process and ask whether it makes sense that Y is a cause of W. This will not convince a skeptical reader, but you should do it anyway. • Models: - Time Order. Necessary but not sufficient. We are on somewhat firmer ground if W precedes Y in time, but the Shalizi & Thomas problem of an as-yet-earlier joint confounder is still there. - Simultaneous Models. Model both the friendship pattern and the outcome of interest simultaneously. Best bet for direct estimation •Sensitivity Analysis: I think the most reasonable solution…take error potential seriously, attempt to evaluate how big a problem it really is. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 180. Table 4. Selected SIENA Parameter Estimates: Parental Knowledge, Parental Discipline, and Drinking a Model 2 b SE t SD Selection parameters Alter effects: Who is more often named as a friend? Parental knowledge -0.002 0.004 -0.47 0.003 Parental discipline -0.004 0.002 -1.55 0.001 Drinking 0.083 0.010 8.69 *** 0.007 Ego effects: Who names more friends? Parental knowledge 0.044 0.007 5.85 *** 0.039 Parental discipline -0.002 0.005 -0.36 0.023 Drinking -0.011 0.021 -0.53 0.089 Similarity effects: Choosing friends similar to oneself Parental knowledge 0.169 0.025 6.70 *** 0.101 Parental discipline 0.151 0.017 8.86 *** 0.035 Drinking 0.276 0.021 13.45 *** 0.006 Behavioral parameters: Influence on Drinking Friends' attributes Mean Parental knowledge -0.230 0.065 -3.56 ** 0.014 Mean Parental discipline -0.051 0.043 -1.18 0.011 Drinking mean similarity 1.162 0.110 10.56 *** 0.023 Control variables (individual level) Parental knowledge -0.122 0.014 -8.93 *** 0.004 Parental discipline -0.043 0.009 -4.54 *** 0.002 ***p < .001. **p < .01. *p < .05. †p < .10. a Models also include rate and shape parameters, structural parameters, and the full set of alter, ego, similarity, and individual-level control parameters SIENA model of drinking Daniel T. Ragan, D. Wayne Osgood
  • 181. Possible solutions: •Sensitivity Analysis: I think the most reasonable solution…take error potential seriously, attempt to evaluate how big a problem it really is. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  • 182. Possible solutions: •Sensitivity Analysis: I think the most reasonable solution…take error potential seriously, attempt to evaluate how big a problem it really is. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Sociological Methods & Research 2000